Striker Ole Gunnar Solskjaer, United captain for the night, had put Alex Ferguson's side ahead before the half hour but Luke Varney scored a deserved equaliser 17 minutes from the end of regulation time.
"There were seven internationals out there for United but you have to work hard before you can win a football match," said Ferguson. "We let Crewe get a toe-hold which we shouldn't have."
Striker Alan Smith, returning to the side for the first time since breaking his leg nine months ago, laid on the winner with a crossfield ball that Lee ran on to to steer past Ben Williams.
England midfielder Joe Cole, also back after a long injury layoff, scored the first goal in Chelsea's 2-0 win at Blackburn Rovers in one of four all-Premier League ties.
Cole struck in the 53rd minute to net a cross from Salomon Kalou, who added the second nine minutes from time off a rebound after keeper Brad Friedel parried Shaun Wright-Phillips' shot.
Newcastle United's Peruvian Nolberto Solano scored twice, his first a fine diving header into the roof of the net from Damien Duff's far post cross, in a 3-0 win over Portsmouth.
Striker Giuseppe Rossi, on loan from Manchester United, had put Newcastle ahead with a turn and low shot in the 48th.
Charlton Athletic, bottom of the Premier League, beat Bolton Wanderers 1-0 at The Valley with striker Marcus Bent heading home in the 17th minute from Jimmy Floyd Hasselbaink's cross.
Liverpool looked to be coasting against Reading at Anfield with a 3-0 lead and 15 minutes to go but in a frantic final quarter of an hour the visitors scored three times although Peter Crouch 77th minute goal ensured the home side won 4-3.
Robbie Fowler, with a fine opening strike, John Arne Riise and Argentine defender Gabriel Paletta, with his first goal for the club, scored Liverpool's first three.
Strike partners Mido and Jermain Defoe scored two apiece to help Tottenham Hotspur rout fourth division MK Dons 5-0. Ireland captain Robbie Keane netted their fifth in the final minute.
The shock of the round was West Ham United's 2-1 defeat at third division Chesterfield on Tuesday - the struggling east London club's eighth consecutive defeat in all competitions.