Opener Kedar Devdhar hit a stroke-filled 145 as Baroda took a sizeable lead of 303 runs, despite narrowly falling short of Mumbai's first innings total, on the third day of their Ranji Trophy Group A match.
Devdhar's knock came of 235 balls that included 16 fours as Baroda finished the day at 321 for five.
With a day's play left, Baroda will try for an outright win as Mumbai have already ensured three points by virtue of 18-run first innings lead.
With Aditya Waghmode (66), Baroda added 177 for opening stand before Deepak Hooda (66 batting, 75 balls) attacked the Mumbai bowlers with six boundaries.
Devdhar and Hooda added 57 runs in 12.4 overs to up the ante. Pathan brothers Irfan (12) and Yusuf Pathan (2) were dismissed cheaply.
The day belonged to Devdhar, who was brilliant and was especially severe on Shardul Thakur (0/57) and Balwinder Singh Sandhu Jr (1/71), treating any width offered with disdain.
Just when Devdhar was looking good, he edged one off Sandhu Jr's bowling.
Baroda: 305 & 321/5 (Kedar Devdhar 145, Deepak Hooda 66, Aditya Waghmode 66).
Mumbai 1st Innings: 323.
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