Tare, who just took over from Suryakumar Yadav, resumed the day at 67 and added another 60 runs to remain unconquered on 127 as Mumbai were all out in 287 in 124 overs.
In fact, Tare scored 60 of the 69 runs Mumbai added in 34 overs today after resuming at 218 for 6. He stitched 50 runs for the last wicket with Akshay Girap (7).
Hardik Pandya top-scored for Baroda with a 91-ball 53 before he was run out while Deepak Hooda was the second highest scorer with 31.
India discards Yusuf and Irfan Pathan scored 9 and 11 respectively as the home side were bundled out for 184 in 54.3 overs.
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Baroda failed to have any big partnership with the highest being the 66 between Pandya and Hooda for the fifth wicket. Pandya hit nine fours in his fine innings.
Mumbai: 287 all out in 124 overs (Aditya Tare 127 not out; Irfan Pathan 3/47, Yusuf Pathan 3/90).
Baroda: 184 all out in 54.3 overs (Hardik Pandya 53; Balwinder Sandhu 4/35, Harmeet Singh 4/52).