Sri Lanka need a further 39 runs to avoid being asked to bat again, still 238 runs behind India's first-innings total of 503-9 declared.
Only 80.4 of the 98 overs scheduled for the day were possible due to rain.
India's fast-bowling coach Morne Morkel said the team intended to wrap up Sri Lanka's innings quickly on day four and enforce the follow-on.
"With the amount of rain that is around, it takes (a) lot of time to put the covers on and take covers off," he told reporters.
"So given the lead we have, we want to ask them to follow on and put them under pressure."
Wicketkeeper Niroshan Dickwella was dismissed for a third-ball duck, reducing Sri Lanka to 173-7.
However, Sonal Dinusha added 57 runs for the eighth wicket with Keshara Nuwantha (18) to frustrate India.
Nuwantha was dismissed soon after tea but Dinusha found another able ally in fast bowler Lahiru Kumara, who batted for more than an hour as the pair added another 35 runs for the ninth wicket.
Dinusha, Sri Lanka's standout performer in the first Test, continued to impress.
The all-rounder, who scored 100 and 84 last week in Galle, ended the day on 85 not out.
Sri Lanka resumed on their overnight score of 8-2 but lost two wickets in the morning session. India made further inroads after lunch, picking up another three wickets.
Pasindu Sooriyabandara, who replaced Dinesh Chandimal as a concussion substitute in the second innings of the Galle Test, is in the starting side for the first time.
Sooriyabandara was the next-best Sri Lankan batsman with 80, but said he regretted not going on to score "a big one."
Prasidh Krishna was the pick of India's bowlers, claiming three wickets, often troubling the batters with extra bounce.
India lead the two-match series 1-0, having won the opening Test in Galle by 165 runs.