The Blades slipped to third as Norwich City won, but they had the chance to respond a day later and, with Leeds United beaten, could have moved within a point of top spot.
However, Chris Wilder's men passed up clear openings in the first half at the Liberty Stadium and were duly punished as Oliver McBurnie netted for Swansea 25 minutes from time.
Billy Sharp and David McGoldrick were both profligate before the break for the Blades, the former shooting too close to Erwin Mulder with the goal gaping before the latter directed a poor header wide.
Swansea was much improved after the restart and got its reward when Nathan Dyer squared for McBurnie to slide the ball into the net.
Dean Henderson made a superb double-save to keep the visitors in the match, brilliantly turning away a curling strike from reported Leeds target Daniel James after blocking an initial shot, but his efforts were in vain.