City had to come from behind three times to earn the draw, with the home side's Moussa Dembele scoring at the start of each half and Raheem Sterling netting a first-half own goal for the Scottish champions.
Celtic were unrecognizable from the side that was humbled 7-0 at Barcelona in their opening game.
But City simply refused to be subdued, responding with strikes from Fernandinho, the excellent Sterling and Nolito and coming close to a late winner that would have equalled Tottenham Hotspur's 11 straight wins in 1960-61 as the best start to a season by any English club.