A goal on the stroke of half time by Kristian O'Leary was enough to give the Swans a 1-0 victory over 10-man Oxford.
The central midfielder showed great agility to jink his way past a couple of defenders before calmly side-footing beyond the advancing keeper.
It moves the Welsh side onto 74 points, with three games of the season to go.
However, they will drop back down to fourth if United beat bottom club Cambridge at Glanford Park.
Three points would also move Scunthorpe above Southend, whose bid to return to the top of the table was dashed by Leyton Orient.
It was the first time the Shrimpers had lost on home turf this year, and marked the Os' first win away from Brisbane Road since November.
Matches between the two sides are always fiercely contested, so it came as no surprise that the game started at a frantic pace, with Fredy Eastwood spurning an early chance.
Orient got the winner when Matt Lockwood's corner from the right was headed in at the far post by John Mackie after 22 minutes.