Amid reports manager Pep Guardiola will step down at the end of the season, City needed to win at Vitality Stadium on Tuesday (Wednesday AEST) to take the title race to the final day.
Erling Haaland equalised in added time to cancel out Junior Kroupi's record-breaking 13th top-flight goal of the campaign, but the visitors were unable to force a winner as Guardiola was denied a potential fairytale farewell.
Bournemouth's historic achievement was a fitting way to mark the final home game of head coach Andoni Iraola, who was given a rousing send-off at fulltime ahead of being replaced by Marco Rose in the summer.
Iraola and Arsenal manager Mikael Arteta have been friends since playing youth football together in San Sebastian.
Following a remarkable 17th league game without loss, the Cherries are guaranteed a top-seven finish and remain in contention for Champions League football ahead of Sunday's season finale at Nottingham Forest.
City are now an insurmountable four points adrift of the table-topping Gunners, with speculation Guardiola is set to end his contract a year early and be succeeded by Enzo Maresca after the weekend visit of Aston Villa.
The 10-time English champions arrived on the south coast three days after lifting the FA Cup but with news of Guardiola's possible departure dominating the headlines.
Travelling fans chanted "one more year, one more year, Guardiola" at regular intervals, while one fan unfurled a bedsheet with 'Pep Stay' written on it.
Kroupi raised the roof with a stunning 39th-minute opener.
After being released by Marcus Tavernier, Adrien Truffert cut the ball back from the left and 19-year-old forward Kroupi curled a stunning finish into the right corner before celebrating becoming the outright highest scoring teenager in a debut Premier League season, with a somersault.
Both Rayan and David Brooks hit the post as Bournemouth threatened to put the match beyond doubt, but in the fourth minute of added time Haaland swept home to jangle to nerves of watching Arsenal fans.
But the home side held on sparking wild celebrations around Arsenal's home ground in London.
Arteta's players can now stand alongside club icons Thierry Henry, Dennis Bergkamp and Ian Wright, who previously led the club to the summit of English soccer. And Arsenal's current class could yet break new ground by winning the Champions League for the first time against defending champion Paris Saint-Germain.
Elsewhere Tottenham lost 2-1 at London rivals Chelsea and remain in danger of relegation.
