Sri Lanka still requires a further 84 runs to avoid an innings defeat.
Resuming from the overnight score of 5-2, Sri Lanka lost night-watchman Dhammika Prasad to the first ball of the day when he hopped and fended a short ball from Varun Aaron to Ajinkya Rahane at gully.
Captain Angelo Mathews and veteran Kumar Sangakkara added 87 runs for the fourth wicket to briefly threaten a fightback, before Sangakkara fell for 40, brilliantly caught one-handed by a diving Rahane at first slip after slashing at a Ravichandran Ashwin delivery.
Sangakkara, playing for the last time in Galle in what is his farewell series, received a standing ovation as he walked off the field. The 37-year-old, who is the leading test run scorer among current players, will end his career after the second test.
Mathews (39) followed Sangakkara back to the pavilion an over later when Lokesh Rahul at silly point snapped up a sharp catch off Amit Mishra.
Mishra and Ashwin bowled exceptionally well and had lot of assistance from the Galle track that had bounce and bite. Both bowlers have taken two wickets apiece.
Lahiru Thirimanne and Dinesh Chandimal had several close calls, but survived through to lunch.
India gained a first innings lead of 192 runs after posting 375 runs in reply to Sri Lanka's first innings total of 183.
Off-spinner Ashwin picked up six wickets in the first innings.
The mainstay of India's innings were captain Virat Kohli and opener Shikhar Dhawan, both of whom made centuries.
India has not won a test series in Sri Lanka since 1993.
Related story
Foreign Minister Paudyal discusses connectivity issue with Sri...
