Jack Wilshere's second-half opener was quickly cancelled out by an Eden Hazard penalty, and Marcos Alonso's strike with six minutes remaining was equalled by Bellerin's late heroics to ensure the two sides shared the points on an entertaining night in London.
The draw sees Chelsea pull within one point of second-placed Manchester United, while Arsenal move a point behind fifth-place Tottenham, who host West Ham on Thursday.
An end-to-end first half was highlighted by key near-misses by each team.
Alvaro Morata shockingly failed to convert a clear chance in the 14th minute. The Spain international found himself all alone behind the Gunners defence. But, with only Petr Cech to beat, he inexplicably scuffed his shot well wide of the post.
Alexis Sanchez then was unlucky not to give Arsenal an immediate answer just two minutes later, but his snap shot hit both posts after an excellent Thibaut Courtois got a hand to his hard-hit effort.
The goals came after the break, however, and it was Wilshere who broke the deadlock just after the hour mark. The reinvigorated England man hooked home from an angle after Granit Xhaka's through ball took a deflection into his path.
But the lead would not last long, and Hazard levelled from the penalty spot four minutes later. Bellerin brought down the Belgian in the box and the Chelsea No. 10 left Cech no chance from 12 yards to ensure the teams would share the spoils on the day.
Chelsea appeared to have completed the comeback in the 84th minute through defender Alonso, who poked home a low cross after a lung-busting run through the middle saw him arrive unmarked in the six-yard box.
However, Bellerin struck with a wonderful volley three minutes into second-half stoppage-time -- moments before another one-on-one miss by Morata -- to earn Arsenal a point on the day.