Despite creating numerous chances, the Blades slipped to a narrow defeat at Bramall Lane.
Moussa Djenepo netted the only goal of the game for Southampton in the second-half, in a game that saw United finish with ten-men, following the late sending-off of Billy Sharp.
For the home encounter against the Saints, boss Wilder made two changes to the team which started at Chelsea, with Fleck returning in place of Luke Freeman and McGoldrick starting ahead of Robinson.
United started positively and forced and early corner, after a free flowing move almost created a chance for an advancing Lundstram. At the other end, Henderson saved comfortably from a header from Vestergaard after Southampton had forced a corner themselves.
Both teams then came close to opening the scoring, within a minute of each other. For Southampton, Adams rattled the post from distance, before the Blades broke swiftly only for Gunn to deny McGoldrick who had been played through on goal by McBurnie.
The game continued to see chances created and again it was the Blades forcing the issue, this time Fleck headed over having timed his run to perfection to meet Baldock's cross. Fleck was involved again moments later, this timing firing an effort wide after Vestergaard misread a diagonal pass from Stevens.
On 26 minutes, United had Henderson to thank for keeping the game goalless, with the Manchester United loanee producing a fantastic reflex save to keep out a downward header by Adams, with Basham denying Højbjerg on the rebound.
Djenepo was the first player to be cautioned for a challenge on Baldock and from the resulting free-kick, United forced another corner from which O'Connell was inches away from making a clean connection.
McBurnie needed lengthy treatment as half-time approached, but the striker was able to return to the action, as Bahsam had to make a perfectly timed tackle to deny Boufal, before the same player saw a shot saved comfortably by Henderson.
Southampton made a few tasty challenges in the opening minutes of the second period and Vestergaard was booked for a late hit on Baldock. And from the resulting free-kick United thought they had gone in front through McBurnie, only for VAR to rule an offside from Norwood's right-wing delivery.
In response, Southampton ventured forward but once again Henderson was equal to an effort from Adams, before the lively Boufal fired over from the edge of the box. In the 65th minute, Wilder made his first change of the game, with Sharp taking the place of the hardworking McBurnie, but within 60 seconds, it was the Saints who went in front as DJENEPO seized on a slip from Norwood before beating three challenges and slotting past Henderson.
The Blades was almost immediate, but from distance, Fleck's rasping drive went inches wide of a scrambling Gunn's far post. Wilder added another attacking option to the pitch with 18 minutes left, as Mousset replaced Lundstram.
As the clocked ticked down, the Blades poured forward, firstly, Fleck was denied by a last-ditch block, before Gunn denied McGoldrick and on the rebound, Mousset fired wide on the turn.
Mousset's pace then created a chance on the break, but Cédric managed to make a needed block.
McGoldrick fired over from a promising position, before substitute Sharp was harshly sent off for a tackle on Armstrong. Six minutes of added time were signalled, which gave United one final hope. The Blades threw everything forward, including Henderson, but the visitors held on.
Blades: Henderson, Baldock, Stevens, Fleck, O'Connell, Basham (Robinson 84), Lundstram (Mousset 72), McBurnie (Sharp 65), Egan, Norwood, McGoldrick.
Unused: Freeman, Osborn, Moore, Bešić.
Southamton: Gunn, Cédric, Vestergaard, Bednarek, Yoshida, Højbjerg, Romeu, Ward-Prowse, Boufal (Armstrong 85), Djenepo (Ings 74), Adams (Long 90).
Unused: McCarthy, Stephens, Bertrand, Valery.
As it happened...
We are underway!
Blades defending during the opening minute. 0-0.
First free-kick of the game goes to the visitors as Jack O'Connell crashes into James Ward-Prowse.
The Blades clear the danger though through George Baldock.
A slow start, but then the game suddenly bursts into life as United put together a flowing move which starts inside the Blades half.
Stevens' low cross is then put behind for a corner which Saints clear.
Now it is Southampton's turn for some flowing football.
The slick move ends with a Ward-Prowse cross which is put behind for a corner.
The subsequent cross sees a Vestergaard header straight into the hands of Henderson.
Big chance for Southampton as Che Adams rattles the post!
Straight up the other end and McBurnie puts McGoldrick clean through on goal but his shot is smothered by the legs of Gunn. 0-0.
The game is coming to the boil now as Fleck's daisy cutter his held by Gunn.
Baldock crosses from the right but Fleck powers his header over the bar. Good spell for United.
What a pass from Stevens cross-field to Fleck.
He chests down before driving into the box. He goes for goal from a tight angle, but a tiny deflection takes the ball wide, and Southampton clear the resulting corner.
Cedric shoots wide after Adams runs over a low pass into the area. 0-0.
What a save by Henderson!
Ward-Prowse with the delivery, Adams rises to head towards goal but the Blades stopper is there to palm away, and Basham makes an important challenge on the rebound.
Corner to United.
Fleck delivers, Saints clear as far as Lundstram who heads back into the area where O'Connell hooks an effort over the bar.
What a pass from Norwood!
Baldock would have been in on goal if he'd have taken it on the run, a difficult one to control and Saints get the ball clear. 0-0.
Fleck free-kick after Baldock was fouled is deflected behind for a corner...
Great delivery, but no-one can get a clean connection. 0-0.
Wonderful free-kick from Norwood into the back post, McGoldrick on the stretch gets something on it but the flag is up for offiside.
Action from the first-half, as John Egan puts in a challenge. 0-0.
Wild effort from McGoldrick goes out for a throw-in on the far side. 0-0.
McBurnie's down having treatment. 0-0 with a minute until half-time.
One minute added time.
Basham make a brilliant challenge to stop a shot on goal, and quickly after Boufal's low effort is easily held by Henderson.
Blades 0-0 Southampton
We're back underway again, 0-0.
First card of the game as Vestergaard is cautioned for a trip on Baldock.
Goal for Blades!!
Check here for VAR...
Goal ruled out for offside. 0-0.
Baldock's cross found McBurnie at the back post but it is ruled out after a VAR check.
McBurnie drills a cross/shot across goal and Southampton clear. Blades on top at this stage. 0-0.
Norwood's deep cross forces a corner on the far side.
The midfielder drills to the back post where Egan heads into a dangerous area but McBurnie can't get the shot away.
Adams shot from distance is palmed away by Henderson.
Mazy run by Boufal ends with a shot high and wide of the far post.
Another quick break by Southampton who are enjoying a good spell. This time a cross field run by Djenepo ends with a weak effort held by Henderson.
Fleck into the book for United
First change sees McBurnie replaced by ex-Saint Billy Sharp.
Goal for Southampton.
Norwood uderhits a pass to Lundstram and Djenepo seizes upon the ball, shrugs off Norwood's challenge and fires beyond Henderson.
Rasping effort from Fleck flashes just wide from 25 yards.
Change for United as Lundstram is replaced by Mousset
Sharp gets in behind the Saints defence, his cross almost finds Basham but the visitors get the ball clear.
Djenepo is replaced by Ings
Shouts for a handball from a United corner. VAR was used, no penalty.
Another shout for a penalty as Basham is challenged, but this was an easier decision as the ball is played.
Three quick-fire chances for United.
First, Fleck has a right-footed effort blocked. The McGoldrick attempts to turn a low cross home but Gunn saves.
Finally, Mousset puts an effort wide. 0-1.
Mousset chases a long clearance and fires in a shot from a tight angle but Cedric makes an important block.
Baldock heads a cross towards the back post into the path of McGoldrick who blazes over from just inside the area.
Final change as Basham is replaced by Robinson.
Saints change, Boufal replaced by Stuart Armstrong
Sharp is sent off for a late tackle on Armstrong
Six minutes added
Adams replaced by Shane Long
Fleck with a dipping cross which almost dropped in, but Gunn is relieved to see the ball drop wide.
Last chance, corner to Blades...
Henderson goes forward and connects with a header but he plants it well wide. 0-1.
Blades 0 - 1 Southampton
Moussa Djenepo's goal wins it for Saints. United had plenty of chances but could not find a way through