Resuming at the overnight score of 85 for four in reply to Tamil Nadu's timid first innings score of 87 all out, Mumbai were bowled out for 176 runs in 67 runs, which gave them an lead of 89 runs.
Skipper Aditya Tare top-scored for Mumbai with a gritty 48-run knock.
Medium pacer Krishnamoorthy Vignesh picked up five wickets for Tamil Nadu giving away just 41 runs from his 18 overs.
At the draw of stumps on the second day, Tamil Nadu has taken an overall lead of 64 runs with four wickets and two full days to play.
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Skipper Abhinav Mukund top-scored with 56 while his opening partner Washington Sundar made 40.
At stumps, Baba Aparajith was batting on 15 while giving him company was J Kousik on 1.
Vijay Gohil (3/36) picked up three wickets for Mumbai, while Dhawal Kulkarani (2/29) accounted for two.
Tamil Nadu: 87 & 153 for 6 in 54 overs (Abhinav Mukund 56, Washington Sundar 40; Vijay Gohil 36).
Mumbai: 176 all out in 67 overs (Aditya Tare 48; Krishnamoorthy Vignesh 5/41).