Brandon Fernandes gave Churchill Brothers the lead from a delectable free-kick in the third minute before Victorino Fernandes (34th minute) levelled before half-time.
Mumbai then won in dramatic fashion as substitute Karan Sawhney -- who missed a penalty earlier in the second half -- scored in the second minute of added time (90+2).
Mumbai have thus won both the matches they have played so far. They are on six points, the same as table toppers and defending champions Bengaluru FC and second placed Mohun Bagan.
But Mumbai have a goal difference of two while Bengaluru and Mohun Bagan have five and three respectively.
Going into the match, the biggest change for the home team was the introduction of Anthony Wolfe who led the Churchill attacking with Brandon Fernandes dropping in just behind him.
The hosts had the best possible start when Pratik Chowdhury brought down Churchill winger Clifford Miranda and conceded a freekick just outside the Mumbai box.
The away side were enjoying more chances and Thoi Singh had two chances in quick succession with a header and a close range shot.
On both occasions, the midfielder was not able to beat Churchill goalkeeper Priyant Singh who thwarted both attempts.
The game was threatening to go off the boil when Mumbai FC were gifted a goal thanks to a glaring error by defender Rowilson Rodrigues.
The defender hesitated with a headed clearance and his half-hearted clearance allowed Victorino Fernandes to pounce on the error and slotted the ball past Priyant to make it 1-1.
In the second half, Mumbai FC were awarded a penalty in the 78th minute when the referee adjudged Churchill defender Adil Khan to have pushed substitute Hitesh Sharma inside the box when they were going for a ball.
Sawhney stepped up to take the penalty, but his weak effort was comfortably saved by Priyant who dived to his right and made the save.
However, Sawhney atoned for his earlier miss when Churchill Brothers were caught sleeping.
Thoi Singh managed to run down the right flank before picking out an unmarked Sawhney in the middle and the rookie guided the ball past the keeper to break hearts of the home side.