Even if the match ends in a draw Thursday, Mumbai, who were 171/1 at stumps, will qualify for the next stage due to their huge first innings lead.
Baroda needed another 175 runs to avoid the follow on and trailed by 374 runs when their innings wrapped up nine minutes before the end of first session. The host skipper Ajit Agarkar decided against imposing the follow on.
First innings centurion Wasim Jaffer became the second player to accumulate more than 9,000 runs and is 97 runs behind former Mumbai player Amol Muzumdar (9,105 runs) in the overall tally.
He scored 33, striking two fours and two sixes, before being trapped LBW by Ambati Rayudu.
Opener Kaustubh Pawar (70) and one down batsman Hiken Shah (65) forged an unbeaten 123-run stand. Baroda skipper Yusuf Pathan seemed to be suffering from butter fingers, as he dropped Hiken Shah (his third dropped catch in the match) on 54. Pawar hit six fours and two sixes, while Shah struck six fours and a six.
Meanwhile, Baroda's leading wicket-taker Murtuja Vahora, who picked four wickets in the first innings including Sachin Tendulkar's, did not come out to bowl in the second innings due to an injury, adding to Baroda's misery.
The visitors, who started the day with 167 runs on board and five wickets in hand, lost Pinal Shah early -- caught in the slip cordon off debutant Vishal Dabholkar.
From then on, it was a downslide for Baroda, as they did not offer much resistance to Mumbai's bowling attack and added only 104 runs in the first session.
The last batsman, Bhargav Bhatt, hit two fours before being caught by Tendulkar off Javed Khan.
Rayudu, dropped earlier on 33, ran out of partners and was left stranded at 89. His knock comprised 15 fours and two sixes.
Dabholkar, playing his first match, was pick of the bowlers as he scalped three wickets for 88 runs and would have bagged Rayudu's wicket too if Agarkar had held onto a simple catch. Javed Khan, Dhawal Kulkarni and Ankeet Chavan grabbed two wickets each.
Mumbai: 645/9 declared and 171/1 (Kaustubh Pawar 70 batting, Hiken Shah 65 batting, Ambati Rayudu 1/32)
Baroda: 271 all out (Ambati Rayudu 89 not out, Aditya Waghmode 53, Saurabh Wakaskar 51, Vishal Dabholkar 3/8