The champions were in control for long spells of the match, but they had to wait until the 55th minute for their only goal as Riyad Mahrez came off the bench to earn City a crucial and hard-fought three points in the Premier League title race.
The result lifts Pep Guardiola's side two points above Liverpool, who now need to beat Everton in Sunday's Merseyside derby at Goodison Park to regain top spot with nine games of the season remaining.
Bournemouth handed a surprise start to defender Jack Simpson, whose only previous Premier League appearance also came against Man City in December 2017, but he was part of a defence which frustrated the dominant visitors during the first half.
Nicolas Otamendi glanced an early header wide of the target from a corner, and it was from set pieces that they looked most like making the breakthrough as they struggled to find a way past the Cherries' rearguard action despite enjoying the lion's share of possession.
City's best chance of the first half did come from open play, though, as Kevin De Bruyne pulled the ball back for David Silva, but the Spaniard could only put his finish wide when he should have hit the target.
Otamendi sent an acrobatic overhead kick over the crossbar shortly after the half-hour mark and Raheem Sterling also came close with a curling effort which drifted just wide, but City in general struggled to turn their territory into chances.
A first half of frustration only grew in stoppage time when De Bruyne pulled up with what looked like a hamstring injury, and Guardiola was then forced into another change in the opening exchanges of the second half when Vincent Kompany came on for John Stones.
The growing injury list would have been a concern for Guardiola, but it was one of the changes who finally gave them the breakthrough 10 minutes into the second half when Mahrez's strike crept under Artur Boruc at the near post after the out-of-form Algerian had been found by Silva.
If Boruc felt that he should have done better with that one, he went some way to redeeming himself less than 10 minutes later by denying Sterling when the Man City winger was clean through on goal.
The Bournemouth keeper was also called upon to save Sergio Aguero's outside-of-the-boot effort before again coming out on top in a one-on-one situation with Sterling, producing a flying intervention to prevent the attacker from rounding him.
Boruc's atonement for his error was completed four minutes later when he got a crucial touch to turn the ball onto the crossbar after the wind had caught Aguero's cross and threatened to help it over the keeper and straight in for a freak goal.
It was Man City who continued to push forward for the game's second goal, though, as Bournemouth failed to register a single shot throughout the 90 minutes compared to 23 for City.
The visitors' 82% possession also hinted at the one-sided nature of the match, and City squandered another chance to reflect that in the scoreline four minutes from time when Boruc produced one last smart save to deny Mahrez's close-range header.
The champions were not made to rue their missed chances as they edged to a second successive 1-0 win - and a fourth league win in a row without conceding - but it was another victory which came at a cost following the injuries to De Bruyne and Stones.
Nevertheless, City now sit two points clear at the top of the table once again and will turn their attention to the Merseyside derby, which Liverpool must win to knock the champions back off the summit.
BOURNEMOUTH (5-4-1): Boruc; Clyne (Rico 83'), Mepham, Simpson (Ibe 83'), Ake, Daniels; Smith, Brooks (Mousset 75'), Surman, Fraser; King
MAN CITY (4-3-3): Ederson; Walker, Stones (Kompany 48'), Otamendi, Zinchenko; De Bruyne (Mahrez 45'), Gundogan, Silva; Bernardo, Aguero (Jesus 90'), Sterling body check tags ::