The Reds almost scored after 70 seconds when Tom Lawrence flashed a long-range effort just wide but, as the hosts warmed to the occasion, Elliott Whitehouse forced a fine, plunging save from Liam Jacob with a 10th minute shot from distance.
Just as the Blades were establishing a foothold in the game, United broke the deadlock. Lawrence's measured ball released Sam Byrne, whose attempt to round George Willis was thwarted, only for on-song Barmby to race in and tuck home the loose ball.
Jacob then beat away a shot from Deacon, before Willis had to drop to his left to turn away a shot from van Velzen. The Dutch winger, operating on the right wing but cutting inside, would not be denied his goal for long, however.
United's lead was almost doubled when Lawrence's curling effort bounced off the crossbar, before van Velzen tapped home the rebound when James Weir's shot was spilled.
As the defences largely bossed the remainder of the game, the closest call for either side came when Whitehouse dragged a shot just wide of Jacob's far post in stoppage time. Paul McGuinness' youngsters registered a first clean sheet of the season and a second successive victory, making it six points from a possible 12 so far this term.
United: Jacob; Hendrie, McCullough, Dalley, Blackett; van Velzen, Rothwell, Weir, Barmby (Daehli, 60); Lawrence, Byrne (McNair, 83). Subs not used: Coll, Pearson, Ekangamene.