It looked like Chelsea had come from behind to triumph when Marcos Alonso struck six minutes from time, after Jack Wilshere's first league goal since May 2015 had been cancelled out by an Eden Hazard penalty.
But the Gunners would not be beaten and Bellerin punished Alonso's poor clearing header with a fine finish for a share of the points - though there was still time for Chelsea's Davide Zappacosta to hit the crossbar.
Wilshere broke the deadlock, but there were doubts over whether the midfielder should have been on the pitch.
Already booked, Wilshere went down looking for a free-kick on the edge of the area and might have received a second yellow card from referee Anthony Taylor for simulation.
The Gunners' lead lasted just four minutes as Hector Bellerin was adjudged to have fouled Hazard, who equalised with the resulting penalty.
Alonso's effort appeared decisive, only for Bellerin to strike.
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