James Tarkowski put Burnley ahead just before half-time, but Irish defender Egan netted with 10 minutes left at Turf Moor.
United remain in eighth place, four points behind sixth placed Wolves, while Burnley move up to ninth, two points adrift of the Blades.
Bucking the trend of many drab first halfs since the restart, there was a lively opening behind closed doors on a blustery afternoon in Lancashire.
Blades striker Oliver McBurnie was put through on goal by David McGoldrick’s, but Burnley keeper Nick Pope made a fine save.
Burnley’s Matej Vydra should have done better than drag his shot wide after being teed up by Erik Pieters.
Sander Berge’s close-range effort was saved by Pope, while the Blades also had a strong penalty appeal for handball by Pieters controversially turned down after a VAR review.
United were dominant and Oliver Norwood’s corner was headed just over by Egan.
McBurnie was inches away from Jack Robinson’s cross, but despite the Blades blitz, it Burnley who took the lead three minutes before half-time.
Tarkowski netted from close-range after Jay Rodriguez headed on Dwight McNeil’s free-kick.
The defender’s second goal of the season had to survive a VAR check for offside before he could celebrate.
Burnley had lost only one of their last 33 Premier League matches when they were leading at half-time.
McGoldrick’s deflected shot forced an agile stop from Pope and the United striker was close again as he stretched in vain to turn in Billy Sharp’s shot.
Jack Rodwell came off the bench for United, making his first Premier League appearance since May 2017 after the former Manchester City and England midfielder’s career stagnated at Sunderland.
Rodriguez brought a good save from Dean Henderson, but McNeil shot wastefully wide from the rebound with just the keeper to beat.
It was a costly miss as United equalised in the 80th minute.
Ben Osborn’s cross was flicked on by Sharp and Egan was left unmarked to fire home with a fine finish at the far post.