They were just over a minute away from victory when a long ball out of defence sent Rovers sub Jo Kuffour on a chase into their penalty area.
Keeper Casper Ankergren had no chance to move as Kuffour smashed in a 20-yard right-footed shot.
Leeds' 2,000 travelling fans who were behind his goal couldn't believe it and neither could United after coming from behind to take the lead in the second half through man-of-the-match Robert Snodgrass.
But Rovers playing with an intensity they have seldom shown at home this season and never gave up hope. They sent on all three subs and, in a highly competitive climax, matched Leeds raid for raid with Darryl Duffy hitting the post with a header.
They had caught Leeds out in the 14th minute when David Pipe swept past sleepy defenders down the right flank. As Manuel Rui Marques ran across in a bid to cut him off, Pipe hit an accurate ball past him and, at the far post, Jeff Hughes fired his shot into the top of the net, the midfielder's sixth strike this season.
Leeds, who had been pressing down the middle trying to create openings for Jermaine Beckford, changed their tactics and, masterminded in midfield by Snodgrass and Jonny Howson, widened their game and stretched the home defence.
It all paid off in the 27th minute when Luciano Becchio rounded off a pass from Snodgrass with Rovers caught out of position.
While Jermaine Beckford failed to add to his 25 goals, he won the free-kick just short of the hour which saw Snodgrass put the Yorkshire side in front, bending a superb 22-yard shot home. It caught Rovers' experienced keeper Steve Phillips defending the wrong post.