Goals from seasoned medio Paresh Shivalkar, who scored off a free-kick, and Brazilian striker Josimar in the first half gave the home side, who were in danger of falling into the relegation zone, the lead.
Ranti Martins pulled a goal back in the second half, which was not enough for the Kolkata side as Khalid Jamil's men ran out creditable winners.
The victory boosted Mumbai FC to the sixth spot with 21 points and a game in hand against league debutantes Bharat FC while East Bengal, who suffered their sixth loss in 18 games, maintained their fourth spot on the table.
Mumbai got more chances to score but Mohd Rafi shot wide and Josimar was foiled in the nick of time. But the home team did not have to wait long for a second goal.
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A through pass by Climax Lawrence enabled Josimar to get into a fine position with only the rival goalkeeper to beat and the Brazilian shot home in the 14th minute to make it 2-0 for Mumbai FC.
East Bengal came closest to score in the 38th minute when Harmanjot Khabra shot inches wide following a pass from Leo Bertos.
Both sides flunked chances to score in the second half and in the end Mumbai FC won.