The visitors impressed with their aggressive approach in the first half and were three goals up at the end of it. Xisco Hernandez (6') opened the scoring before goals from Aridane Santana (21', 73') and Jerry Mawihmingthanga (41') sealed the win.
Mohamed Larbi's (51') penalty and a late fortuitous hit by Bipin Singh (90+4') turned out to be nothing more than consolation for Mumbai who looked lacklustre.
Odisha got off to a terrific start and scored in the sixth minute. Santana played the ball towards Xisco who dribbled into the box, dropped his shoulder to beat Pratik Chaudhari and curled a simple finish into the net.
Josep Gombau's men were then denied by the crossbar as they chased a second goal. Santana's deflected effort from the edge of the box ricocheted wide off the woodwork.
But the visitors soon doubled their lead midway into the first half. Shubham Sarangi released Jerry down the right flank whose low cross to the edge of the box was turned in by Santana with an excellent first-time effort.
Mumbai weren't without chances at the other end, but lacked composure inside the box. Amine Chermiti failed to hit the target from a Serge Kevyn cross and Diego Carlos' cross for Kevyn went wide.
The hosts' defence was punished with another blow just before the break. Nandhakumar Sekar cut in from the left and sent a curling effort at the goal which hit the bar. The rebound fell to Jerry who took a touch and dispatched it into the net to stun the home crowd.
Mumbai's pursuit for a way back into the game was helped when Kevyn chased a lob into the box and collided with an onrushing Francisco Dorronsoro, forcing the referee to point to the spot. Mohamed Larbi calmly slotted home from 12 yards to open a window of hope for his team.
The game slowed down considerably after the hour-mark and Mumbai struggled to sustain the intensity with which they started the second half. Odisha waited for their moment and soon added the fourth goal.
( With inputs from IANS )