The hosts, replying to India's first innings score of 312, slipped to 11-2 before going to lunch on the third day at the Sinhalese Sports Club on a shaky 47-5.
Pujara marked his recall to the Test side with an unbeaten 145 as the tourists added 20 runs to their overnight score of 292-8 before being dismissed 25 minutes after the start of the day's play.
Sri Lanka lost both openers with the score on 11 as Upul Tharanga was snapped up at second slip by Lokesh Rahul off Ishant Sharma and Kaushal Silva was bowled by Umesh Yadav.
Dinesh Chandimal looked in fine touch, smashing five boundaries in his 23, before he was unlucky to be given out lbw to Stuart Binny. Television replays indicated the ball would have gone over the stumps.
Sharma dealt a major blow by having skipper Angelo Mathews caught behind for one, giving debutant wicket-keeper Naman Ojha his first dismissal.
Dimuth Karunaratne was dismissed off the last ball before lunch for 11, caught in the slips off Binny, to leave Lahiru Thirimanne unbeaten on zero.
Pujara, who had been dropped for India's last four Tests across Australia, Bangladesh and the ongoing series, may not have played this match if both regular openers Murali Vijay and Shikhar Dhawan had been fit.
The 27-year-old, whose usual place in the batting order is at number three, notched up his seventh Test century and become only the fourth Indian after Sunil Gavaskar, Rahul Dravid and Virender Sehwag to bat through the innings.
Left-arm spinner Rangana Herath bowled the last two batsmen, Sharma and Yadav, to finish with three wickets for 84 runs.
But fast bowler Dhammika Prasad was the most impressive Sri Lankan bowler with four wickets for 100 runs on a greenish pitch that favoured seam bowling.
The series is locked at 1-1 after Sri Lanka won the opening Test in Galle by 63 runs and India drew level with a 278-run win in the second match at the P. Sara Oval in Colombo on Monday.