After being bowled out for 290 in their first innings, Sri Lanka appeared to be clawing their way back into the match through defiant knocks from Pasindu Sooriyabandara (92) and Kamindu Mendis (55).
However, once the pair were separated, Sri Lanka's middle order stuttered again. They finished day four on 229-6, 16 runs ahead, with only four wickets in hand.
Sonal Dinusha, who top-scored with a fluent 103 in Sri Lanka's first innings, was unbeaten on seven, with Keshara Nuwantha on three when rain stopped play yet again.
Earlier, debutant spinner Saransh Jain claimed the final two Sri Lankan wickets, prompting India captain Shubman Gill to enforce the follow-on.
Mohammed Siraj removed Lahiru Udara in the fourth over, and Ravindra Jadeja dismissed Kamil Mishara for 32.
Sooriyabandara and Mendis, however, launched a counter-attack, scoring freely during a 115-run stand for the third wicket.
Sooriyabandara had a slice of luck when a diving Siraj spilled a difficult chance at deep square leg. On another occasion, the ball fell just short of Jadeja before Gill produced a brilliant catch to break the partnership.
Mendis top-edged a short ball from Prasidh Krishna and Gill, positioned at square leg, first ran backwards and then sideways before diving to hold on to the swirling ball.
Sooriyabandara, who made 80 in the first innings of only his second test, looked set for a maiden hundred after successfully reviewing an lbw decision that was overturned.
But the batter then suffered a momentary lapse in concentration. He danced down the track to a wide delivery from spinner Manav Suthar, only to miss it completely. India wicketkeeper Rishabh Pant stretched fully to gather the ball, but had ample time to complete a straightforward stumping.