Karnataka have already secured a 189-run lead over their opponents with five wickets still remaining here at the Wankhede stadium.
Rahul (131) and Nair (130) were unconquered at the end of day's play, which saw them overhaul Tamil Nadu's first innings paltry total of 134.
Resuming at an overnight score of 45 for four, title the holders lost night watchman Abhimanyu Mithun (39) early to be reeling at 84 for five.
The 25-year-old hit 16 off the second over of the day, hitting Laxmipathy Balaji for three boundaries.
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Mithun's overnight partner, Karun Nair, having resumed at nine, carried on to notch up his first century of the season.
He added 239 runs with Rahul for an unbeaten partnership for the sixth wicket.
The Indian Test player Rahul hit 13 fours and a six to bring up his century. Rahul, who had retired hurt yesterday due to a hamstring pull after scoring one, batted patiently with Nair to equalise Tamil Nadu's first innings score of 134 at lunch.
Nair, who came in to bat yesterday at number six, was slower than Rahul in bringing his century. He took 248 balls to reach the three-digit mark compared to Rahul's 170. He struck 19 fours in his knock.
The Karnataka batsmen applied themselves to restrict Tamil Nadu from picking any further wickets.
After the fall of 14 wickets yesterday, just one batsman was dismissed today. It wasn't a bad day at the office for the Tamil Nadu bowlers.