At stumps on the penultimate day, Baroda were 116 for four with Ambati Rayudu and Pinal Shah at the crease.
Tamil Nadu had scored 230 in their first innings, thereby taking a 22-run lead over the hosts who were bowled out for 208 in their first innings.
Ramaswamy Prasanna top-scored for Tamil Nadu with a defiant knock of 75 off 213 balls, while Baba Aparajith made an equally tenacious 62 off 191 at the Moti Bagh Stadium.
Prasanna, in all, spent 315 minutes at the crease, and Aparajith was there for 272 minutes.
Murtuja Vahora was the most successful bowler for Baroda, returning impressive figures of 4/36, and there were two wickets apiece for Sankalp Vohra, Gagandeep Singh and Ketan Panchal.
When Baroda came out to bat for the second time in the match, they were well served by Aditya Waghmode (39), but Jagannathan Kaushik struck twice late in the day to leave the hosts at 105 for four.
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Brief scores:
Baroda: 208 and 116 for four in 66 overs (Aditya Waghmode 39; Jagannathan Kaushik 3/23)
Tamil Nadu 1st innings: 230 all out in 102.3 overs (Ramaswamy Prasanna 75, Baba Aparajith 62; Murtuja Vahora 4/36).