Even second-half substitutes Wayne Rooney and Cristiano Ronaldo could not break Burton's superb resistance as Nigel Clough's men pulled off the biggest result in the club's 55-year history.
Clough admitted his total surprise at the result but hailed a "heroic performance".
He said: "We had a little bit of luck in the second half but over the 90 minutes I thought the lads deserved the chance to go to Old Trafford. We had a couple of half-chances ourselves and we thought we had a decent shout for a penalty."
United fielded a much-weakened team and it looked as though Sir Alex Ferguson had underestimated the Brewers.
They were the better team in the first half and twice had United on the rack.
In the 14th minute Jon Shaw rose to meet a Keith Gilroy corner but saw his header cleared off the line by Phillip Bardsley.
Skipper Darren Stride suffered a similar fate a minute before half-time, with Bardsley again the man on crucial goal-line duty for United.
The introduction of Ronaldo and Rooney just before the hour mark inevitably saw United start to show their Premiership class, but the pair still could not find a way through a resilient defence.
Justice was done in stoppage-time when Saul Deeney saved brilliantly with his legs to deny United youngster Richie Jones, after Louis Saha had headed down a Mikael Silvestre free-kick.
Sir Alex Ferguson was disappointed not to have won, but admitted Burton deserved their money-spinning replay.
"They are satisfied with the draw and to come to Old Trafford - they are still in the Cup," he said.
"We were expected to win. But football is not as easy as that - you've seen the nature of FA Cup football today. Anything can happen.
"It was a difficult pitch, and our opponents were committed - you can get shocks, and fortunately we didn't get a severe shock today.
"Nothing really serious happened against us apart from a couple of corners which were scrambled clear in the first half.
"We had some half-chances and one or two good chances. But I won't deny Burton - it's a great draw for them."
Man of the Match
Burton: Darren Tinson
The combined talents of Louis Saha, Wayne Rooney and Ole-Gunnar Solskjaer could not get the better of Albion's defensive rock.
Manchester United: Phil Bardsley
The youngster pushed his claims for a regular starting role with two crucial goalline clearances and some adventurous surges forward.