The visitors took the lead at a nervy Emirates through Junior Kroupi but Mikel Arteta’s men levelled through a Viktor Gyokeres penalty later in the first half.
The Gunners made multiple attacking changes early in the second period but struggled to create clear-cut chances against Andoni Iraola’s enterprising team.
Bournemouth were back in front in the 74th minute after a fluid move finished off by Alex Scott, who rifled the ball past David Raya.
Arsenal pushed for a leveller but slumped to just their second home defeat in the league this season.
With six games to go, Arsenal, chasing their first Premier League title since 2004, are nine points clear of second-placed City.
But Guardiola’s men can close to within three points of the top if they beat Chelsea on Sunday and overcome the Gunners in an Etihad summit meeting next week.
Arsenal came into the match lifted by their last-gasp 1-0 win in the first leg of their Champions League quarter-final against Sporting Lisbon in midweek.
But they were sluggish against a bright Bournemouth team who played the more fluid football in the opening stages.
Iraola’s team took a deserved lead in the 17th minute when Kroupi tapped in from close range after the ball fell to him following a wicked lopping deflection off William Saliba.
The goal followed a well-worked move, with Bournemouth captain Ryan Christie playing a sweetly measured ball into the path of Adrien Truffert, whose cross was deflected into his path by Saliba.
Kroupi, 19, is the first teenager to score 10 goals in his debut Premier League season since Robbie Keane for Coventry City in 1999/2000.
Moments later Kai Havertz squandered a clear chance to level, putting a header over the bar and the crowd became increasingly edgy as the home side struggled to settle.
But Arsenal were level in the 35th minute when Gyokeres blasted home from the penalty spot after the ball struck Christie’s outstretched hand.
Arteta, who has been criticised for being over-cautious this season, made three attacking changes early in the second half, bringing on Eberechi Eze, 16-year-old Max Dowman and Leandro Trossard for Havertz, Noni Madueke and Gabriel Martinelli.
But the changes failed to lift the Gunners, who looked short of attacking ideas beyond long balls to Gyokeres, who fired wide in added time when well placed.
Just weeks ago Arsenal were on track for a unique quadruple before defeat to City in the League Cup final and a shock loss to Southampton in the FA Cup.
They remain favourites for the Premier League title but City, with two games in hand, will travel to Stamford Bridge with renewed hope.
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