Championship leaders Burnley needed a 90th-minute goal from Connor Roberts to edge out the 10 men of League One Fleetwood 1-0 and advance to the quarter-finals of the FA Cup for the first time in 20 years.
incent Kompany will now return 'home' to face his former club Manchester City after the quarter-final draw was made.
Burnley played with an extra man for more than 45 minutes after Cian Hayes saw red in first-half stoppage time, but were in danger of being taken to extra time by a Fleetwood side who showed why they had come into this match on the back of four straight clean sheets.
When the goal did come it was harsh on visiting goalkeeper Jay Lynch, who made fine saves from Halil Dervisoglu and Charlie Taylor in quick succession only to be beaten from close range as Roberts' flick diverted Vitinho's shot through a crowded area into the net.
It was reward for a Burnley performance that had seen them dominate possession all night and register 31 attempts at goal against a side playing in the fifth round for the first time.
The hosts were almost gifted an opener eight minutes in when an under pressure Aristote Nsiala went to pass the ball back but did not check where Lynch was, sending the goalkeeper scrambling to hook it off the line.
That proved to be as close as Burnley got to a goal before the break as Scott Brown's men made them work.
Lyle Foster should have done better when he robbed Harrison Holgate of the ball on the halfway line and ran free towards goal. But the South African, still looking for his first Burnley goal, got his finish wrong as he chipped into the hands of a grateful Lynch.
The Burnley pressure was building as a couple of shots were charged down before the lively Vitinho blazed over, while Taylor had a pop from range but sent it high and wide before testing Lynch with a better effort the goalkeeper needed two opportunities to gather.
The ball then came to Anass Zaroury, who quickly controlled and got a shot away to force Lynch into a save to his right at the expense of a corner.
Fleetwood, barely seen as an attacking force, had a couple of opportunities from free-kicks late in the half but failed to create anything from either.
And things then changed in first-half stoppage time as Hayes crashed into Josh Cullen when challenging for a high ball, with Dean Whitestone's decision to show a red card appearing harsh on second viewing.
Reduced in number, Fleetwood retreated further into their shell after the break, packing the defence with bodies and frustrating Burnley's attempts to find a way through.
Zaroury flashed a shot into the side-netting in the 56th minute, then Lynch did well to deny Foster as he reacted quickest to a loose ball.
Burnley frustration grew as Scott Twine hit a free-kick into the wall after Scott Robertson fouled Cullen in the D.
Nathan Tella, whose brace saw off Ipswich in the previous round, was one of three players to come on just after the hour. Roberts, another of the fresh faces, fired in a low cross for Tella, but his shot was blocked at the near post.
Moments later, Lynch got a hand to Taylor's header, taking just enough pace off it that Holgate could hook it off the line.
Lynch then clawed away shots from Dervisoglu and Taylor, but when the ball came back into the box, Roberts got the decisive flick.
FA Cup quarter-final draw
Man City v Burnley
Man Utd v Fulham
Brighton v Grimsby
Sheffield Utd v Blackburn
The ties will be played over the weekend of Saturday March 18