Third-placed East Bengal's win was made possible owing to a thunderous long-range volley from Cavin Lobo on the hour mark at the Tau Devi Lal Stadium.
Apart from the early few minutes, East Bengal dominated the game having penetrated Minerva's defence with sharp through balls from the flanks.
Minerva Punjabs goalkeeper Rakshit had to come out of his position numerous times to clear the ball from one-on-one situation to keep the scoreline unscathed.
Meanwhile, East Bengal missed an easy chance to go forward when Yami Longvah failed to connect a cross inside the box in the 35th minute, resulting in an early change by Khalid Jamil, who introduced Ghenafo Kromah at his expense.
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At half time, it was even-stevens as both teams tried adjusting to the ground conditions.
Minerva then had another chance when William was tripped in a dangerous position just outside the penalty box but the resultant free kick by Sukhdev was off-target.
But an absolute stunner by Lobo silenced the home crowd to give his side the lead with 30 minutes of game-time left.
Chencho had an extremely good chance to level the score in the 79th minute but his shot went straight into the goalie's gloves.
The visitors then held on their nerves for the remaining minutes to get three crucial points and stay alive in the title race.
The Kolkata giants will now travel to Kozhikode to challenge Gokulam Kerala FC on February 17, while Minerva Punjab FC will host the Kerala-based side three days later.
Minerva Punjab have 29 points from 14 games, second-placed Neroca FC have 28 from 15 matches, while East Bengal have 26 points from 14.