Paulo Machado's (15') free-kick goal was cancelled out by Tiri's (37') header from a corner kick. Raynier (56') then came off the bench to produce the match-winning goal.
Jamshedpur and Mumbai City are now level on the table with 13 points from nine games, with the former occupying the fourth spot owing to a superior goal difference.
The first major chance of the game fell to Mumbai as early as the third minute. Bipin Singh attempted an audacious volley off a Modou Sougou cross from the right flank. The strike took a deflection on its way towards the goal and forced a good save from Subrata Paul.
The Islanders were persistent and converted the next chance that they got. Paulo Machado delivered a stunning free kick from the left flank which beat Paul at his near-post.
Jamshedpur sought a way back into the game and grew in confidence as the half wore on. C.K. Vineeth nicked the ball off Machado from midfield, dribbled towards the edge of the box and had his shot blocked. The rebound fell to Piti whose ferocious strike flew inches wide of Amrinder Singh's goal.
Minutes later, a terrific through-ball by Tiri split the Mumbai defence and fed Farukh Choudhary into the box, but the India international shot wide with just the keeper at his mercy.
The hosts, however, equalised from a corner-kick in the 37th minute. Tiri outfoxed his marker and leapt the highest inside the box to beat Amrinder with a close-range header.
After what was an even start to the second half, half-time substitute Raynier regained Mumbai's lead in the 56th minute. Machado won the ball in midfield and passed it forward to Sougou, who held off his marker with his back to the goal. The Senegalese striker laid the ball off for Raynier, who unleashed a magnificent first-time effort from outside the box that flew into the net.
After going behind, Jamshedpur switched gears again and Amrinder was soon called into action at the other end. Farukh chested a chip into the box from Isaac Vanmalsawma onto the path of Vineeth whose snapshot was kept out brilliantly by the Mumbai custodian.
Jamshedpur continued to pressurise the visitors as the game wore on and fashioned a few half-chances to equalise. But a gallant Mumbai defence held on to seal an important win.
( With inputs from IANS )