Resuming at the overnight score of 332 for 2, the home side lost two wickets including that of centurion M S Washington Sundar and finished the day at 357 for 4 in 88 overs, taking a lead of 99 runs over Tripura in the first innings.
The left-handed opener fell for the addition of just three runs to his overnight score of 156, caught by Yashpal Singh off Abhijit Sarkar.
He hit one four before becoming Sarkar's second wicket.
Test star Ravichandran Ashwin joined Baba Indrajith (89 batting) in the middle and a few deliveries later, the rains arrived and play had to be stopped.
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A combination of persistent rain and bad light meant an early end to the day's play.
The truncated third day would affect Tamil Nadu's hopes of an outright victory.
The team had conceded the first innings lead in the drawn encounter against Andhra Pradesh in the opening round.
(Smit Patel 99, K Vignesh 4 for 41) vs Tamil Nadu 357 for 4 in 88 overs (M S Washington Sundar 159, B Indrajith 89 batting, Abhinav Mukund 76, Abhijit Sarkar 2 for 43).
At Vadodara: Baroda 373 all out in 128 overs (Swapnil Singh 88, A Sheth 70 not out, Vishnu Solanki 61, B Ayyappa 4 for 69) vs Andhra Pradesh 505 for 9 in 139.4 overs (G Hanuma Vihari 150, Ricky Bhui 145, A Sheth 5 for 77 ).
At Indore: Madhya Pradesh 409 in 144.3 overs