Minerva's teenage stars Baoringdao Bodo (57th minute) and Anirudh Thapa (66th) struck in the second half to dash Santosh Kashyap's hopes to defeat the debutants.
Even though Karan Sawhney (77th) minimised the gap with his third goal of the season, it couldn't deter the Punjab outfit from clinching their maiden win in the country's premier football league.
After a goalless first-half, the match livened up in the second half, with both teams eager to break the deadlock.
Minerva managed to score their first goal after a deflected cross from Anirudh Thapa found Bodo at the right place at the right time.
The next one came in quick succession for the home team after Thapa scored directly from a free-kick on the left flank, which landed on the goal beating goalkeeper Laxmikant Kattimani on the far post.
Mumbai became more attacking and it was all down to Steven Dias' in-swinger which was headed on to goal by Sawhney, beating Arnab Das, the custodian.
With the win, Minerva leapfrogged Chennai City FC to the ninth spot in the table with five points whereas Mumbai FC are still searching for the elusive point following five consecutive defeats in the league.