Sukhdev Singh put Minerva ahead in the 19th minute, scoring from a deflection off East Bengal goalkeeper CK Ubaid at the Barasat Stadium.
The visitors doubled the lead with Bhutanese 'sniper' Chencho Gyeltshen fooled three East Bengal defenders for a calm and composed finish.
East Bengal's tale of missed chances continued as their star Japanese Katsumi Yusa failed to convert a penalty in the 50th minute with Minerva's goalkeeper Kiran Limbu pulling off a fine save to be adjudged hero-of-the-match.
Meanwhile, coming off from a four-match winning run in away matches, the Punjab outfit dropped points for the second time as they are now on 26 points from 11 matches.
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East Bengal now have 20 points from 12 matches and remain put at third place with six games left. The result comes as a big blow to their maiden I-League title aspiration.
Minerva were rock solid in their defence and that was the reason why even the likes of Yusa found it difficult to penetrate.
Minutes later, Sukhdev got the ball from Kamalpreet Singh's corner as his shot returned to him from a deflection off Ubaid and he finally found the back of the net to give the lead in an unconvincing fashion.
East Bengal's defence went from being bad to worse as there was utter chaos in their box as Armand Bazie and Eduardo clashed with each other letting Chencho find the back of the net enroute his sixth goal of the season.
East Bengal also seemed unlucky to have been denied a penalty in the in the first-half injury time when Yusa was brought down inside the box by Guy Eric Dano.
In the 50th minute, Laldanmawia Ralte was brought down by Kassim Aidara inside the box but it hardly mattered as Yusa failed to convert from the ensuing spotkick.
East Bengal then went on to dish out an exciting last half an hour's play beginning with Yusa's perfect corner with Justin finding a flawless finish to pull one back.
In full throttle, East Bengal looked to go ahead but were unfortunate as Ralte's 90+3rd minute shot came off from the upright.
Minerva also had their chances from a counter attack but the ball went over the bar as the Punjab outfit ended their frustrating away campaign, dropping points for only the second time in 11 matches.