A goal from a homegrown, Sheffield youngster earned the Blades a slender lead in the first leg of the play-off semi-final.
Substitute Callum McFadzean scored the only goal of the night at Bramall Lane, albeit with the aid of a deflection.
United defended solidly to set up an intriguing second leg in Somerset some 66 hours after the conclusion of the first leg.
Robson and Porter returned to the starting XI for the first leg, the experienced duo taking over from McFadzean and Ironside, who both dropped to the bench.
Caretaker chief Morgan kicked off with Robson on the right flank and Murphy on the left, supporting the powerful front two of Porter and Kitson. In a frantic opening, the early chances fell to United, beginning with Kitson mis-timing a header from Robson's cross on five minutes.
Robson fired a free-kick just wide after he had been fouled by Upson and then, after a penalty appeal had been turned down when Stech appeared to man-handle McDonald, Porter headed a Robson corner wide.
Like they did in the league game at the Lane, Yeovil pressured the Blades defence but the opportunities continued for the home side and Murphy fired over on 25 minutes after both Maguire and Collins both got on the end of a Robson free-kick.
The Blades lost Robson to injury just after the half hour, McFadzean taking over, and the home side picked up the game's first booking, Kitson penalised for a trip. Yeovil's first legitimate shot on goal came from Dawson, who fired across Long and wide, whilst it was followed by a breakaway attempt ending with Edwards seeing a shot from a dangerous position blocked.
Just a minute had elapsed in the second half when the Blades seized the advantage, from a move down the left. Murphy went around the outside of his marker and his low cut backed cross picked out McFADZEAN, whose first time effort took a decisive deflection off McAllister and found the net, via Stech's right post.
Yeovil hit back with two chances of their own, Upson's long-ranger moved in the air but was kept out by Long, with Maguire clearing up the danger, and then Upson provided the pass to open up the Blades defence by Hayter was called offside as he lifted into the net.
Doyle was the second Blade in the book and around the hour Murphy made tracks across the box but his shot was straight at Stech. McFadzean was impressing and backed himself to beat Yeovil skipper McAllister but the subsequent shot, on 68 minutes, was palmed away by the visiting goalkeeper.
McFadzean's next contribution, after McDonald won possession, was to find Murphy, who fired wide, but Yeovil went very close on 73 minutes, Madden meeting Ayling's cross but Collins headed to safety over his own crossbar.
With 15 minutes to go Ironside's fresh legs took over from Kitson but United were forced to repel a number of Yeovil efforts, including Webster's header from Upson's corner, which was well saved and held by Long.
Ironside almost got on the end of McDonald's chip forward after Stech needlessly conceded a corner, whilst the Yeovil fans were appealing for a penalty on 88 minutes when Maguire kneed the ball against his own arm.
Four minutes of added time were announced and Long was forced into another save, pushing Webster's acrobatic effort away, whilst late on Maguire was booked for reasons unknown and Whitehouse replaced Porter.
Blades: Long, Westlake, Maguire, Hill, Doyle, Porter (Whitehouse 90), McDonald, Kitson (Ironside 75), Collins, Robson (McFadzean 31), Murphy.
Unused: Higginbotham, Chapell, Barry, Coyne.
Yeovil: Stech, Ayling, McAllister, Webster, Upson, Hayter (Balanta 73), Foley, Madden, Burn, Edwards, Dawson.
Unused: Hinds, Blizzard, Williams, Stewart, Maksimenko, Young.