A second half brace from Scottish striker Darryl Duffy (66th and 72nd minute), and a late goal by winger Gilbert Oliviera (90th) handed the visitors three points.
Substitute Jayesh Rane pulled one back for the hosts in the 85th minute.
Following the defeat Mumbai FC stay in fifth place with six points from five games, while Salgaocar go top of the table with ten points from four.
Coach Khalid Jamil started with same team that faced United SC in the previous game, with strikers Sandjar Ahmadi and Orok Essien starting upfront.
Duffy, who was later named man of the match, was the first on to go close and also get booked for simulation after diving inside the box.
At the other end, Mumbai FC had striker Sandjar go wide from distance. The hosts kept pressing and saw medio Climax Lawerence’s effort off a free-kick on the edge of the box go just over.
Midway through the half winger Ashutosh Mehta's perfect cross into the box from the right was agonisingly headed over by striker Orok. Minutes later, defender Haroon Amiri’s curling shot directly off the corner on the left almost sneaked in at Salgaocar’s far post.
Mumbai FC’s dominance in the first-half failed to produce a goal and both teams went into the break tied at goalless.
Salgaocar turned up the heat in the second half and took the lead six minutes past the hour mark. Left-back Rahul Kumar sent striker Duffy in the clear on the left and he cut past a defender inside the box before hitting a grounder, which beat Mumbai FC custodian Kunal Sawant.
Six minutes later, Duffy completed his brace as he headed home Gilbert’s floated cross into the box from the left past the Mumbai FC ‘keeper.
The hosts pulled one back five minutes from the final whistle to set up a nervy finish. Sandjar’s header off Orok’s cross in from the left was saved by Salgaocar ‘keeper Karanjit Singh. However substitute striker Jayesh -- on for Kamaljeet Kumar -- timed his run to perfection and slotted home at the far post.
Any hopes of a Mumbai FC comeback were quashed in the final minute of normal time as the visitors scored their third goal. Gilbert controlled Duffy’s crisp through ball inside the box and kept his cool to round off Sawant and score.
Image: Action from the I-League match between Salgaocar (green jersey) and Mumbai FC
Photograph: AIFF Media
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