The draw leaves United winless and with just two points after two matches, while Basel is top of the group with four along with Benfica, which beat newcomer Otelul Galati 1-0.
"We were careless.. it's a wake-up call in many ways," United manager Alex Ferguson said. "A lack of concentration saw us give the game away really ... but we rescued it."
There was no comeback in Munich for Manchester City as Mario Gomez twice knocked in rebounds from Manchester City goalkeeper Joe Hart to put Bayern 2-0 up before half time.
The victory left Bayern in control of Group A with a perfect record after two games, while City has one point as well as having to deal with insubordination.
City manager Roberto Mancini wanted to bring Tevez off the bench in the second half as City was trying to recover from 2-0 down, but the Argentina striker refused to take the field.
"If I have my way he will be out of the club," Mancini said, while stressing that the club's Abu Dhabi owners would make the final decision. Despite being the Premier League's joint top-scorer last season, the striker has featured infrequently this campaign after failing to secure a transfer away from Manchester in the offseason.
City is being kept off the bottom of Group A by Villarreal, which has no points after losing 2-0 at Napoli.
After Marek Hamsik's 14th-minute opener, Edinson Cavani doubled the lead from the penalty spot three minutes later.
Real Madrid enjoyed the biggest victory of the night to stay top of Group D and leave Ajax on one point.
AP