McBurnie came up with the game’s only goal in the 65th minute as the Blades missed the opportunity to make up ground on Leeds at the top of the Championship.
Chris Wilder’s men dominated the first half, but couldn’t convert their chances and McBurnie showed them how it should be done.
Leeds’ surprise defeat at Stoke opened things up for the teams behind them, but United couldn’t take advantage as strikers Billy Sharp and David McGoldrick had an off-night in front of goal.
Swansea captain Mike Van Der Hoorn took an early blow to the head but continued as United dominated the first half.
John Fleck had a shot from distance and George Baldock fired over the crossbar with Swansea yet to get going.
Graham Potter’s side did lose Joe Rodon to a foot injury and he was replaced by Cameron Carter-Vickers.
The Blades should have taken the lead when Enda Stevens crossed from the left. Visiting captain Sharp side-footed towards goal, but Erwin Mulder produced a smart low save.
Swansea were dicing with death and after Oliver Norwood was booked for sending Connor Roberts flying, United then missed another golden opportunity.
This time it was Mark Duffy’s cross from the right which was headed wide by McGoldrick when it looked easier to score.
Bersant Celina had Swansea’s first effort on goal before the break, but there was no doubting his team was second best.
Potter replaced Wayne Routledge with Dan James for the start of the second half and the young winger, who is reportedly interesting Leeds, had a big impact.
Matt Grimes curled a free-kick over the bar and Swansea were better after the break as they at last carried an attacking threat.
McBurnie poked wide from a Nathan Dyer ball before then coming up trumps.
From a set-piece, Dyer played a neat one-two with Van der Hoorn and the former passed the ball across the six-yard box for McBurnie who couldn’t miss.
Swansea, all of a sudden, were purring.
United goalkeeper Dean Henderson produced two stunning saves from Celina and then James to keep it at 1-0. The second of those in particular was an acrobatic effort.
Swansea may have failed to double their advantage, but in the end they did enough to hold on thanks to McBurnie.
United pushed hard in the final 10 minutes and went direct in search of an equaliser by winning a series of corners, but this was not their night as Sharp appealed for a penalty for handball against Van der Hoorn in four minutes of added time.