Man of the Match Omar El Hussieny converted from the spot at the halfway mark, while Dipanda Dicka completed the tally in the 69th minute on his comeback match.
The win kept Mohun Bagan's slender title hopes alive as they remained sixth on the table with 18 points from 12 matches, while Minerva Punjab are a spot below them with 13 points from as many games.
Mohun Bagan next face NEROCA in three days' time, before the return leg derby on January 27.
Jamil made as many as five changes from the side that suffered a 1-2 loss to Real Kashmir.
The I-League winning former Aizawl FC coach preferred Dipanda Dicka over Henry Kisekka, and he also brought in Azharuddin Mallick, Darren Caldeira, Gurjinder Kumar and Dalraj Singh in the starting line-up.
The new-look Mohun Bagan side appeared rejuvenated under Jamil as the home side completely dominated the first half at the Salt Lake Stadium, and went ahead through a penalty.
Minerva missed the services of William Opoku big time as their attack lacked bite.
They had no no shot at goal in the first half and struggled to keep the possession.
Mohun Bagan continued their domination in the second half as well with Yuta Kinowaki and Omar winning the midfield battle against Minerva.
Sony Norde set up the second ball with a sublime through ball that bisected the two centre-backs before reaching Dipanda.
The Cameroonian forward kept his head down and calmly shot the ball with only Roy to beat in front of him.
In the 74th minute, Mohun Bagan could have increased their tally but Roy dived to his left to deny Omar after the Egyptian was brilliantly set up by Yuta Kinowaki just at the edge of the box.
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