Led by Baba Inderjit after regular captain and opener Abhinav Mukund pulled out of the game at the MCA-BKC ground due to indisposition, TN lost India Test opener Murali Vijay cheaply before recovering to 104 for 1.
But Dabholkar struck thrice in quick succession to reduce Tamil Nadu, who were asked to bat first, to 107 for 4 before the visitors ended the day at 249 for six by riding on a responsible, unbeaten knock of 76 by Karthik.
At stumps, Karthik and Malolan Rangarajan (12 not out) were holding fort.
For Mumbai Dabhokar, who was chosen instead of Harmeet Singh who was unimpressive in the innings win over Punjab in Mumbai's previous game, ended the day with impressive figures of 4 for 67.
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Mumbai medium pacers Shardul Thakur, who sent back Vijay for 8 caught behind with only 20 on board, and Balwinder Singh Sandhu took the other two wickets to fall.
Tamil Nadu 249 for 6 (B Aparajith 62, K Gandhi 31, Dinesh Karthik not out 76, J Koushik 39, M Rangarajan not out 12; V Dabholkar 4 for 67).