Sarkar struck in the first (90+1) and third minute (90+3) of added time at the Tilak Maidan Stadium.
Dawda Ceesay had scored a first-half opener before a brace from Sarkar sealed three points for Indian Arrows.
Match-winner Sarkar did not start the game and Raheem Ali was omitted due to an injury.
Kalu Ogba came into the lineup in place of Bektur Talgat in what was the only change made by Alfred Fernandes to his team.
Naorem threated in the eighth minute when he picked up Tangri's pass. His curler forced a very good save from James Kithan who had to dive to his left to keep the ball out.
Despite the early threat from Arrows, it was Churchill who broke the deadlock through Ceesay. Nicholas Fernandes surged forward with the ball and delivered an accurate cross into the box from the left flank, which was easily nodded in by an unmarked Ceesay.
As the match entered the injury time Indian Arrows shocked the local supporters by scoring two quick goals in two minutes.
Sarkar equalised with a right-footed shot from the centre of the box to the top right corner off a Borish Thangjam assist. He then scored the all-important goal two minutes later.
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