Che Adams grabbed the winner and Billy Sharp added his 20th goal of the season as United made it four games unbeaten.
In a battling performance at Shrewsbury, United kept the slim hopes of reaching the play-offs alive.
For a fourth consecutive game, manager Adkins named an unchanged starting XI and bench. Coutts and Adams created an early shooting opportunity for Sharp but the majority of the danger in the opening 10 minutes came at the opposite end of the pitch.
Vassell's precise ball through the defence gave Akpa Akpro a clear pathway to goal but, bizarrely, he stood still, maybe thinking he was offside, allowing United to hack away before Shrews were literally handed a great chance when McEveley handled Kaikai's cross in the box. The skipper was booked but from the resulting penalty Long saved low to his left from Vassell and then a combination of the goalkeeper, Flynn and McEveley crowded out the taker and Akpa Akpro in the race for the rebound.
Sharp almost attached himself to an Adams shot soon after but the leading scorer was just moments away from his 20th of the campaign. It was a neat, well-weighted pass from Flynn down the right channel and SHARP did the rest, firing past Leutwiler before he could react.
To their credit, Shrewsbury were quickly back on the offensive and after a clash of feet which left Baptiste on the deck and Akpa Akpro in the clear, McEveley was quickly across to make a block.
Long was the busier of the goalkeepers in the first half hour but the remainder of the first half was a more even encounter with United possibly edging the attacking endeavour, with Flynn particularly prominent.
In the opening 90 seconds after the interval, Shrewsbury had a penalty shout, a corner and forced a goalmouth scramble but the Blades survived the onslaught, ahead of an assistant referee pulling up with an injury - it took a couple of minutes for the fourth official to get himself ready to take over with the flag.
Shrewsbury were soon back on the attack and levelled the scores soon after the restart, although 54 minutes were on the clock. Kaikai burst into the box and his pass looked behind OGOGO but he curled his foot around the ball and beat Long.
The home side's good start to the second half continued with Akpa Akpro heading over a cross from Sadler cross and Kaikai's shot being deflected wide. Their efforts, though, were hit by the Blades retaking the lead on 66 minutes. ADAMS played a neat one-two with Sharp to find space and a fortunate deflection took the ball away from Leutwiler and into the corner of the net.
Shrewsbury created two quick opportunities to try and drag themselves level again but Akpa Akpro missed the target and Mangan headed over. As the clock ticked round to 80 minutes, Akpa Akpro was free at the back post to connect with Kaikai's cross but Long was well-positioned to save.
Expectedly, the late pressure came from Shrewsbury as they looked for points to secure their League One position. Baptiste blocked from Kaikai and all across the back line important headers and clearances were made.
Due to the assistant referee change, seven minutes of added time was announced. On the break, Cuvelier thought he should have had a penalty following Whitbread challenge and then the home side's final chance saw Kaikai fire a free-kick into the wall.
Shrewsbury: Leutwiler, Sadler, Whitbread, Vassell (Mangan 65), Kaikai, Wesolowski, Grimmer, Ogogo, Knight-Percival, Hendry (Whalley 79), Akpa Akpro.
Unused: Halstead, Black, Mangan, Clark, Cole, Smith.
Blades: Long, Basham, Flynn (Cuvelier 85), Coutts, Sharp, Done, McEveley, Baptiste, Adams (Woolford 90), Edgar, Hammond.
Unused: Howard, Sammon, Calvert-Lewin, McGahey, Whiteman.