The Mumbai outfit enjoyed a slender 1-0 half-time advantage at the Cooperage football ground.
After failing to score in 180 minutes of play in their first two games, Mumbai FC finally ended the goal drought when Japanese midfielder Taisuke blasted home the opening goal in the 37th minute.
Later Sushil Kumar Singh did well to put the finishing touches from a Taisuke assist in the 48th minute.
In the 85th minute, Salgocar nearly got the equaliser, but Darryl Duffy's header off a Gurjinder Singh free-kick graced the upright and sailed over.
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Both teams showed little urgency at the start.
But midway through the first session, Mumbai FC coach Khalid Jamil's decision to switch Eric Brown with Jayesh Rane on the left midfield helped their attack come to life.
The hosts messed up a few chances with Izumi Arata and Taisuke missing the target.
On the stroke of the half-time, Mumbai FC goal survived a scare when goalkeeper Pawan Kumar fumbled to collect Gurjinder Singh's free-kick from the right. The ball deflected off his hands and struck the crossbar. He again failed to collect cleanly but defender Min-Chol Son, who had a good game, cleared the ball away for a corner.
Jackichand pulled one back for the Goans when he sent a long chip from the right which eluded the outstretched hands of Mumbai goalkeeper Pawan Kumar and dipped into the far corner of the net.
Veteran Dias started the move, this time from inside
his own half, as he put Brown in possession near the half line and the latter smartly evaded a couple of tackles from the rival defenders before setting up Sushil who chased the ball and fired to the top right corner giving Paul no chance to make a save.
The second half was more or less on even lines. But only during this period the Pune outfit showed more urgency but could create much of an impact.
Mumbai FC struck the fourth and last goal through Brown in the 90th minute. The Liberian was later adjudged 'Man-of-the-match' and received a cheque of Rs 10,000.