Replying to Sri Lanka's first innings 305, Latham and Devon Conway gave New Zealand a solid start on Thursday by putting on 63 for the first wicket before the opener then added 73 for the second wicket with Williamson.
Latham looked comfortable against the spinners until he fell on 70. Attempting to sweep Prabath Jayasuriya, the ball came off his top edge and he was caught at square leg.
Williamson batted with authority and was on 55 when a lapse of concentration saw him glove a Dhananjaya de Silva delivery for wicketkeeper Kusal Mendis to take a terrific diving catch.
De Silva also bowled Rachin Ravindra for 39 when he failed to read the spinner's straight ball and shouldered arms.
An unbroken 59-run stand for the fifth wicket between Daryl Mitchell (41) and Tom Blundell (18) then took New Zealand to stumps.
Latham said: "It's been a great day for us. Superb effort by the bowlers to get three wickets early and then the partnerships have been able to get us to that total. We put their spinners under pressure."
Earlier, New Zealand did well to clean up the Sri Lankan tail. The last four wickets fell for three runs, fast bowler Will O'Rourke finishing with 5-55.
The morning session was ruined by rain as only 4.5 overs were possible. Play will get underway 15 minutes early on day three.
Craig Howard, Sri Lanka's spin bowling coach, said: "We felt we were in a very good position yesterday. Ideally we would have liked to get another 40 runs today. If we had bowled like we generally do, we could have been better off."
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