Ben Brereton Diaz sent the visitors ahead from the penalty spot in the 21st minute of the contest before an own goal from Sam Gallagher levelled the scores seven minutes later.
Sammie Szmodics's 60th-minute effort saw Rovers lead 2-1 on the hour, but Oli McBurnie made it 2-2 in the 81st minute before Tommy Doyle's stunner won it for Sheffield United in stoppage time.
The Blades are four-time winners of the FA Cup, and they are now just one game away from the final of this season's competition.
Sheffield United had the first chance of the match in the sixth minute when John Egan broke out of defence before finding James McAtee with a through ball, but the midfielder's effort was saved by Aynsley Pears.
Blackburn's goalkeeper was again called into action one minute later to deny Doyle, with the home side making a fast start to the contest at Bramall Lane.
Sheffield United's Wes Foderingham then made a brilliant stop in the 18th minute to deny Gallagher, who had seen his effort deflect off Jack Robinson and towards the back of the net.
Blackburn were awarded a penalty in the 20th minute for a Robinson handball, and Brereton Diaz stepped forward to send Rovers ahead with a strike into the bottom corner.
The Blades were only behind for seven minutes, though, with Max Lowe's strike taking a huge deflection off Gallagher before dropping past Pears into the back of the net.
Foderingham made an excellent save to keep out a header from Dominic Hyam in the 33rd minute, with the two teams entering the half-time interval on terms.
The first half was full of incidents, and it was also a brilliant start to the second period, with McAtee missing the post with a low effort before George Baldock hit the side-netting moments later.
Foderingham then made two excellent saves in quick succession, first keeping out Harry Pickering's strike before Brereton Diaz's rebound was also stopped by the Sheffield United goalkeeper.
McAtee again came close just before the hour, with the midfielder denied by Pears, but the game's third goal arrived on the hour, with Szmodics expertly finding the bottom corner after being released by Gallagher, as Blackburn took a 2-1 lead at Bramall Lane.
Ryan Hedges hit the post for Blackburn in the 71st minute, with Rovers threatening a third, but Sheffield United were level 10 minutes later, with McBurnie finding the bottom corner past Pears after smart work from Lowe.
The match looked to be heading for extra-time, but Sheffield United had other ideas, with Doyle finding the top corner with a sensational strike in the first minute of added time at the end of the 90 to send the Blades into the final four of the competition.