United, Cup winners on a record 11 occasions and runners-up last season, fielded only a handful of first team regulars and looked distinctly ordinary on a sandy, bumpy pitch against a spirited Conference side of part-timers.
Elsewhere, Tottenham Hotspur, fourth in the Premier League, blew a 2-0 lead to lose 3-2 at second division strugglers Leicester City while fourth division Leyton Orient pulled off the shock of the round when they overcame Premier League Fulham 2-1 at Craven Cottage.
Sunderland, bottom of the top flight, beat minor league Northwich Victoria 3-0.
A year ago a weakened Manchester United were held 0-0 at home by minor league Exeter City and had to recall the big names to secure a 2-0 victory in the replay.
This season, despite being out of Europe and with little hope of catching Chelsea in the league, Ferguson repeated his tactic with only Wes Brown, Mikael Silvestre and John O'Shea from his usual first-choice side starting the match.
The unfamiliar lineup, with striker Ole Gunnar Solskjaer making his first start for 19 months, struggled to impose themselves on a hard-working Burton side, managed by Nigel Clough, son of the late former Nottingham Forest manager Brian.
United did not really look dangerous until the introduction of Wayne Rooney and Cristiano Ronaldo for the last 30 minutes but their best chance came in injury time when goalkeeper Saul Deeney saved with his feet from Ritchie Jones.
"I had nothing to do really all game which was a bit of a surprise against one of the best teams in the world," Deeney told Sky Sports.
Clough said: "It was a staggering performance really. In the first half it was pretty even but in the second it seemed like we were under the cosh for about
"Now the lads have got a couple of days off to get ready for Stourbridge on Tuesday night. But I'm thrilled for them that they will get the chance to run out at Old Trafford."
Ferguson said: "Everybody in the country would expect us to beat them but that's cup football.
"We're great at keeping teams alive. Last year we gave Exeter a big pay day, this year it's Burton.
"It was disappointing in many ways, but at the end of the day I'm satisfied to get the draw and we're still in the Cup."
Orient led 2-0 at halftime after goals by Craig Easton and Joe Keith but had to survive a strong second-half comeback by their London rivals.
Collins John pulled a goal back five minutes after the restart but then blew the chance to save the game when his 69th-minute penalty was well saved by Orient keeper Glyn Garner.
Tottenham also led 2-0 after Jermaine Jenas scored in the 20th minute and Paul Stalteri slammed in a 20-metre screamer in the 41st.
However, Leicester, fourth-last in the second division, replied through Elvis Hammond just before halftime and levelled with a powerful shot by Stephen Hughes in the 57th minute.
Spurs, knocked out of the League Cup by Grimsby earlier in the season, completed a miserable double when Mark de Vries broke clear to tuck in the winner in injury time.
Several other top flight sides face replays after being held by lower-league opposition on Saturday. Middlesbrough drew 1-1 with minor league Nuneaton Borough, Birmingham City were held goalless at fourth division Torquay, Everton drew 1-1 at second division Millwall while Wigan Athletic also drew 1-1 with second division Leeds United.
European champions Liverpool came from 3-1 down to win 5-3 at second division Luton Town while Cup holders Arsenal beat second division Cardiff City 2-1.
The fourth round draw will be made on Monday.