Chandimal was unbeaten on 44 and Mathews on 16 as Sri Lanka reached 184-3 at tea, chasing Zimbabwe's 356.
Chandimal and Mathews, who quit the captaincy this week after Sri Lanka's loss in the one-day series against Zimbabwe, put on 68 for the fourth wicket.
Opener Upul Tharanga gave Sri Lanka a quick start before being run out at the non-strikers end as the hosts slipped from 84-0 to 116-3 at the R Premadasa Stadium in Colombo.
Tiripano's lucky fielding effort came after he got Zimbabwe's first breakthrough in Dimuth Karunaratne (25) in his first over after lunch. Karunaratne's wicket ended an 84- run opening partnership.
Zimbabwe skipper Graeme Cremer got Kusal Mendis caught behind for 11 with his off-spin.
Earlier Zimbabwe lasted just 4.4 overs of the morning session, adding 12 runs to their overnight score of 344-8, before the innings ended.
Craig Ervine top-scored with 160 while Sri Lankan spinner Rangana Herath returned an impressive 5-116.
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