Pankaj Singh took 4 for 41 for Rajasthan. S Badrinath (50), Ramaswamy Prasanna (56 batting) and Vijay Shankar (57 batting) hit half-centuries for Tamil Nadu.
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.
Unheralded Ramaswamy Prasanna scored his first century in five years before skipper Laxmipathy Balaji bowled a fiery opening spell to rip apart the Mumbai top-order as Tamil Nadu look firm favourites to enter the Ranji Trophy final.