Achille Emana (22') gave the Indian Super League (ISL) side the lead while Katsumi Yusa (26') equalised soon after. Mahmoud Al Amna (73') finished matters for the I-League outfit in the second half to set up a quarter-final date with Aizawl FC.
Aiborlang Khongjee returned after a long injury lay-off, making his first appearance of the season. Lucian Goian had the armband while Achille Emana was the other of the only two foreigners that Mumbai City had available.
East Bengal's first game under the Subash Bhowmick-Khalid Jamil era had Khalid Aucho making his debut in midfield with Dudu Omagbemi, who had scored seven goals in his last three games, leading the line.
Both Mumbai City midfielders, Zakeer Mundampara and Sehnaj Singh were booked within the first 10 minutes. Amrinder Singh had to pull off a quick collection of saves in the eleventh minute as his defenders failed to clear a ball into the box from Katsumi Yusa towards Dudu.
In spite of the yellow card, Sehnaj Singh continued to play with fire as he kept tackling Katsumi dangerously. The referee was reluctant to raise the red card despite some silly fouls from the Punjab-born midfielder.
Eduardo Ferreira brought down Sahil Tavora and Emana stepped up to take the free-kick. The Cameroonian's stunning shot from around 35 yards went over the wall and crashed into the net off the post to give the ISL side the lead.
A reply beckoned four minutes later courtesy Katsumi Yusa. Laldanmawia Ralte broke free and delivered a wonderful cross from the right. The Japanese playmaker headed home, meeting the ball near the farpost as Amrinder failed to track the ball.
Katsumi provided another moment of brilliance seven minutes after the change of ends when he swept the ball off Mundampara's feet. The former Mohun Bagan man drove inside the box with urgency and lashed a powerful shot which Amrinder managed to save.
Cavin Lobo played a lovely ball over the top to find Ralte's run on the right flank. Amrinder parried away the winger's shot but Al Amna, stationed at the near post, blasted the ball into the centre of the net to score the winner.
Emana's sporadic efforts to gain parity went in vain as the Red and Golds saw out the remaining quarter of an hour comfortably. They now play Jamil's former side Aizawl FC in the first quarter-final on Sunday.