Haaland rose highest in the 50th minute to side-foot past Emiliano Martinez for his 10th goal in six Premier League matches, equalling the league's record for the fewest amount of games to reach double figures. Bailey, though, responded in the 74th minute to earn Villa an impressive point against the reigning champions.
Midfielder Jacob Ramsey was particularly lively for the hosts at Villa Park, providing the pass for his side's equaliser and causing Pep Guardiola's side several problems.
City failed to click into gear at first and it was Villa who arguably had the biggest chance of the first half when Ollie Watkins was sent through on goal in the 38th minute and fired a shot wide of Ederson's left-hand post after a tackle from John Stones.
But it did not take long for City to assert their superiority after the break, with Haaland appearing at the back post to volley home after a looping cross from Kevin De Bruyne.
Villa had a chance to equalise in the 58th minute when Ramsey won possession and burst on to Watkins' return pass, but a poor touch from the midfielder allowed Ederson to come out and smother.
De Bruyne almost put City further ahead with a free-kick which skimmed the top of the crossbar in the 67th minute, while Haaland worked Martinez shortly afterwards with a right-footed effort inside the penalty area.
But Villa burst into life with 16 minutes remaining when Ramsey darted forward and produced a low cross for Bailey, who swept home to draw the sides level.
No VAR review was possible for that effort, but replays suggested Coutinho may have been in an onside position and Villa boss Steven Gerrard said it had been a "legitimate goal".
"Good game," Guardiola said. "We conceded just one goal from one shot on target in 90 minutes.
"We were not precise in our final-third touches and our simple things, especially in the first half. We had chances to score a second when the game was in our hands but we didn't."
Villa are in 17th and have four points from their opening six games but Gerrard suggested he had been rewarded for a more pragmatic approach.
"Very proud -- the players have come together today," Gerrard said. "They put an incredible amount of effort into the game, followed instructions.
"City are always going to have moments and we had to ride our luck, this is what we wanted out of the game and this is the game plan we gave them."
Information from Reuters contributed to this report.