Starting on their overnight score of 317 for three at the Reliance Stadium, Mumbai lost Iyer (173) early as the top-order batsman was run out by Yusuf Pathan after adding just 6 runs to his Day 1 score.
Surakumar Yadav (68) could also add just 2 runs to his overnight score as the Mumbai innings started wobbling. But Abhishk Nayar struck a defiant 89 and stitched a 70-run tenth-wicket stand with Shardul Thakur (24) to give muscle to the Mumbai score.
In reply, the hosts lost their opener early after Monil Patel (10) was scalped by Dhawal Kulkarni for 10, but Kedar Devdhar (84 not out) and captain Aditya Waghmode (35 not out) stitched an unbeaten 113-run stand to give the home team hope.
They still trail by 314 runs in the first innings.
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Brief Scores:
Mumbai 447 all out in 127.1 overs (Shreyas Iyer 173, Suryakumar Yadav 68, Abhishk Nayar 89; Ajitesh Argal 4-67, Hardik Pandya 2-50)
Baroda 133/1 in 45 overs (Kedar Devdhar 84 not out, Aditya Waghmode 35 not out; Dhawal Kulkarni 1-31).