Skipper Angelo Mathews was unbeaten on 110 and Kusal Perera was dismissed just before tea for 70 after they had added 135 for the sixth wicket at the Sinhalese Sports Club.
The hosts will resume after tea needing 137 more runs to win in a possible 36 overs with four wickets in hand.
The series is locked at 1-1, with Sri Lanka winning the opening Test in Galle by 63 runs and India drawing level with a 278-run victory in the second match at the P. Sara Oval in Colombo.