At the P Sara Oval, Mathews (72 batting) and Thirimanne (57 batting) batted without any problems as they added 84 runs in the 28 overs bowled. They still trail India by 169 runs.
Starting at the overnight score of 140/3, India's main hope was in inducing mistakes from the batsmen before they got set.
And it almost worked as Ishant Sharma (0-38) got Thirimanne to edge one on the fifth ball of the morning.
It could be rated as a half-chance, and that was the only one on offer for the rest of session as the two batsmen played grinding knocks.
Skipper Virat Kohli changed around his bowlers regularly, and to their credit they bowled with control and the run-rate never crossed three-per-over.
But it wasn't enough as Umesh Yadav (1-55), Stuart Binny (0-36), R Ashwin (1-63) and Amit Mishra (1-26) toiled without reward.
The pacers beat the batsmen on ample occasions but that one good chance was missing.
Unhindered on a helpful wicket, the two batsmen put on 110 unbeaten runs for the fourth wicket then.
Their 50-partnership came up off 92 balls, and then Mathews celebrated his 23rd Test half-century off 81 balls.
The Lankan 200-mark came up in the 73rd over and soon after they celebrated their 100-partnership coming off 210 balls.
Thirimanne too reached his fourth Test fifty, off 142 balls, as India took the new ball immediately after the completion of 80 overs.
On day one, KL Rahul scored 108 runs, his second Test hundred, and put up 164 runs with Virat Kohli (78) to help India score 300-plus in one day's play.
Rohit Sharma also scored 79 runs as the visitors had recovered from 12/2 at one stage after winning the toss and opting to bat.
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