Strikes from Sergio Aguero and Kevin de Bruyne twice gave City the lead in Manuel Pellgrini's farewell home game.
But Olivier Giroud and Alexis Sanchez goals mean new boss Pep Guardiola could take charge of a side outside European football's elite club competition.
Arsenal need a point against Aston Villa to clinch their top-four place.
Analysis: Why fifth place is not a disaster for Man City
A draw at home to the already relegated Midlands side would mean Champions League qualification for the 20th successive season for Arsene Wenger's men, who could yet overhaul their local rivals Tottenham for second.