In the pouring South Yorkshire rain, both sides were evenly matched in the first half, with Ruben Neves coming closest to scoring for Wolves when one of his trademark free kicks hit the crossbar and went over the top.
Veteran Sheffield United captain Billy Sharp did have the ball in the net early in the second half, but he was ruled offside, and George Baldock timed his run into the penalty area to perfection but headed just wide.
Wolves never really hit top gear and did not have a shot on target in the second half, but they looked to have done enough to secure a point.
However, deep into stoppage time, Egan climbed highest to head home the winner from a corner and spark jubilant scenes on the Blades' bench.
A second home win since the return of the Premier League moved Sheffield United up to seventh in the standings, above Arsenal, and within a point of Wolves in sixth.
Looking to close the gap on Manchester United in fifth place, Wolves struggled to create any openings of note, and followed their home defeat by Arsenal with another loss.
A win for Manchester United at Aston Villa on Thursday would leave Wolves six points off fifth place - a position that could be good enough to qualify for next season's Champions League, if Manchester City fail to overturn their two-year European ban.
"We have to stay focused in the last minute," Wolves manager Nuno Espirito Santo said. "We had good movement and good chances that we should be more clinical on.
"It's been a tough period but we have to raise our standards and go again."
European football is still a possibility for Sheffield United, who made it three home Premier League wins in a row for the first time since 1993.
"We were up against one of the stand-out teams in the Premier League so I am delighted with how we played," Sheffield United manager Chris Wilder said.
"I would have been satisfied with draw but to get the win is incredible. We cannot look too far ahead, as we should not be where we are. We just have to look at the next game and only that."