Openers Siddarth Saraf (63) and Vineet Saxena (83) provided the base and then Ashok Menaria (62) and Hrishikesh Kanitkar (38 batting) capitalised to bring the visitors to 307 for four wickets at draw of stumps.
Rajasthan now need just 11 runs to surpass Tamil Nadu first innings total of 318.
Saraf and Saxena added 163 runs before the former made exit. Eight overs later, Saxena also walked back and Rajasthan also suffered another blow when a dependable Robin Bist was clean bowled by Rahil Shah for 3.
Menaria made a superb 62 off 69 balls with 11 fours and two sixes.
At close, Rajesh Bishnoi (20) was giving company to Kanitkar.