The Blades started on the front foot and opened the scoring in the 12th minute. After a succession of early corners, COCKERLINE forced the ball home from another excellent Whiteman centre.
The lead lasted barely 90 seconds though, as Swansea equalised from a corner of their own at the other end, a goal that was assisted by poor United marking.
The young Swans probably shaded the remainder of the first-half in terms of possession but the best chances fell the way of the home side.
Heh was instrumental in creating an opportunity for himself, before his pressing saw Cockerline and Ishmail have sights of goal but it was all square at the break.
Reed replaced the returning McDonagh at half-time, as the Northern Irish winger made his comeback after eight months out with a double leg break and it was the introduction of United's youngster ever Football League player who helped turn the game in the Blades favour.
In the 55th minute confusion between Evans and Eastwood allowed Swansea to take the lead, but United's response to falling behind was impressive.
Only excellent defending from the visitors kept the Blades at bay from a flurry of corners, but they were powerless to stop REED who smashed home from 25 yards in the 71st minute after being teed up by Whiteman.
Ten minutes later, after maintaining the pressure, United went ahead. Ptak stormed down the left and Heh was only denied by a terrific save, but from the resulting corner EYRE nipped across his marker to head home Reed's inviting centre.
Eyre rattled the bar moments later, before Eastwood's assured handling ensured the Blades would net a welcome win.
Coach Travis Binnion remarked: "It was a really good performance, particularly the second-half where we dominated possession and forced Swansea into mistakes.
"It was nice for everyone to see Jamie McDonagh back, he has worked incredibly hard for eight months and did very well in the first-half.
"All in all it was a very pleasing performance, the schoolboys were excellent and the more senior lads were first-class too."
Blades: Eastwood, Ramsey (Eyre 64), Ptak, Evans, Cheeseman, Whiteman, Heh, Smith (Hutchinson 77), Ishamil, Cockerline, McDonagh (Reed 45).