It was a shootout disaster for City who failed to find the net once with Rovers winning 3-0 in the dramatic penalty ending.
Both sides also scored from the spot during extra-time with Rovers' coming in the 118th minute to force the penalty shootout.
It was really no more than the lowly League One side deserved for a tremendous battling performance in which they matched their Premier League opponents stride for stride for most of the game.
City looked dangerous at the start as they sprayed the ball about and it needed a fine save from Andy Warrington to prevent Antoine Sibierski from scoring in the 13th minute.
Yasser Hussein was also denied by Warrington three minutes later but once Rovers settled to their task they enjoyed a purple patch with City hard pressed to keep them out.
Paul Green sent a diving header over the top and Rovers had appeals for a penalty waved aside when Lewis Guy was sent tumbling in the box by Ben Thatcher.
It needed a desperate save with his legs from David James to keep out Michael McIndoe in the 21st minute and Mark Albrighton headed just over the top before the interval.
There was little to choose between the sides in the second half with Sean McDaid having a 20-yard shot saved by James in the 50th minute before Warrington did well at the other end to stop a dipping effort from Stephen Ireland.
Sun Jihai hit the bar in stoppage time and the escape allowed Rovers to take the game into extra-time.
Darius Vassell fired City in front from the penalty spot in the 95th minute after Phil McGuire was adjudged to have brought down Richard Dunne.
But Doncaster kept battling although they looked to be in trouble until Warrington made a brave save at the feet of Nedum Onuoha in the 105th minute.
Both players collided, with Warrington in agony and Onuoha was shown the red card by the referee.
The Rovers keeper was stretchered off with Budtz coming on to become the hero in the shootout.
But it needed a penalty in the 118th minute from McIndoe after Paul Heffernan had been brought down to take the game to the limit.
Budtz distinguished himself in the last minute of the game with a fine save from Vassell and took the honours again the shootout.
McIndoe, James Coppinger and Heffernan converted the first three Doncaster penalties, while Vassell hit the bar and then Budtz saved brilliantly from Sibierski and Dunne to give Rovers victory and send the home fans wild with delight.