The hosts, who had lost four of their last five ties, received a major boost as Joyner Lourenco returned to the central defence while Sehnaj Singh partnered Raynier Fernandes at the centre of the park.
Sergio Lobera also rang in changes from their last win against Chennaiyin FC, as Mandar Desai, Brandon Fernandes and Carlos Pena reclaimed their starting berths.
The first 15 minutes of the match was mostly scrappy, with both teams being able to only create a half-chance apiece. Goa's quick counter involved picturesque passing by Coro, Jacki and Brandon before Goian's timely challenge. At the other end, Issoko's effort was deflected for a corner, after Machado found him with a long diagonal cross.
Mumbai took the lead in a similar fashion on the 20th minute, when Arnold once again bamboozled Mandar on the flanks with his pace to send an out-footed cross to the far post, where Rafael Bastos was lurking unmarked and tapped the ball home to earn his side the lead.
It took the travellers ten minutes to equalize as Coro's long-ranger was thwarted unconvincingly by Amrinder and Jacki was at hand to hit the back of the net from hand-shaking distance.
On the 39th minute, Brandon's corner was inadvertently flicked by Edu onto the far post where Mourtada Fall reached the ball before Subhasish and nodded it home to earn his team the lead.
Mumbai had a brilliant chance to equalize in a couple of minutes, but Naveen Kumar pulled off a sublime save and parried away Sougou's powerful close-ranger.
There was still drama left in the first half, as Naveen fumbled while gathering a routine ball and Sougou bundled it home, only for the linesman to raise his flag for outside correctly.
Goa made it 3-1 on the 50th minute and Coro finally opened his account in the knock-outs after scoring 33 league stage goals over two seasons. Seriton played the ball to Jacki whose cross was nodded home by the former Espanyol striker.
When it rained goals, it poured. Brandon floated in a corner into dangerous territory and Mourtada beat his teammate Edu to complete his brace.
On the 67th minute, Serinton rounded a defender off inside the penalty box and hit the post while Lenny's 25-yard flyer was tipped over the bar.
The hosts looked hapless and depended on set-pieces but Goian's off-target header from almost touching distance of the target didn't help their cause.
Brandon put the final nail in the coffin when he latched onto a Hugo Boumous pass with defenders caught napping and calmly beat Amrinder at the near post to make the score 5-1.
The two teams will meet in Goa on Tuesday, with Mumbai requiring a miracle to deny Goa their deserved spot in the final slated for next weekend.