Burnley had looked on the way to three points after Jay Rodriguez's 90th-minute strike put them ahead for the second time in the match.
The Clarets raced into a two-goal lead early in the second half after goals from Dean Marney and Chris Eagles, but they were countered by Daniel Bogdanovic and Ched Evans as a 2-2 draw looked likely.
But the game exploded late on with Rodriguez's strike, which was cancelled out four minutes into injury time by Yeates' low effort, topping off a man-of-the-match display.
The Blades made an encouraging start, almost taking the lead inside the opening five minutes. Bogdanovic was set free down the left and he rolled an inviting square pass to Johnny Ertl, whose thumping effort from the edge of the area was tipped over by Burnley keeper Brian Jensen.
Burnley should have taken the lead just before the half-hour when Andre Bikey inexplicably headed over from three yards out when he had the goal at his mercy following Dan Fox's corner.
The game had suddenly burst into life and United were again denied by a good Jensen save when he palmed away Stephen Quinn's low effort.
And soon after Jensen was beaten, but the woodwork came to his and Burnley's rescue when Yeates' curling free-kick hit the outside of the post and went out to safety.
The outstanding Yeates brought another save out of Jensen as the Blades started the second period on the front foot, but their ascendancy did not last long as Burnley took a 49th-minute lead.
Eagles and Mears worked the ball well down the right and the former Manchester United winger curled in a cross which was headed home by Marney, who celebrated his first goal for the club.
And it got even better for the Clarets five minutes later when Myers' searching run saw him hauled down in the box by Ertl, and Eagles sent the resulting penalty home in resounding fashion.
United had to launch a quick response if they were to get anything out of the game and they were handed a way back into the game just three minutes later.
Mears was looking comfortable in possession before losing his footing and allowing Bogdanovic to steal in and race clear to tuck home his first goal for the club.
United's task should have been made easier just after the hour, but Burnley's Fox unbelievably escaped a red card after a shocking late tackle on Ertl.
However, Speed's men found a leveller of their own accord when Yeates' pull back was converted by Evans with five minutes remaining.
It looked as though Burnley were not to be denied the win when Rodriguez stole in to plant a low finish in the bottom corner in the last minute.
But United were not to be denied and they dug deep to find a leveller when Yeates struck from 18 yards from Stephen Jordan's cross.