Openers Tare (64 off 50 balls) and Shoaib Shaikh (54) shared a 102-run stand, setting up an ideal platform to chase down Baroda's 175 for 7.
Suryakumar Yadav (16*) and Safaraz Khan (14*) took their team home on the final ball of the 20th over.
Earlier, Mumbai let Baroda put up a competitive total after opting to field.
Saurabh Wakaskar (35), Yusuf Pathan (33) and Krunal Pandya (51) were the main contributors to the score.
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