Dananjaya had 5-50 in 18.2 overs at Galle.
New Zealand resumed on 71-3 after lunch and Ross Tayor and Henry Nicolls added 100 runs for the fourth wicket. But New Zealand lost two quick wickets close to the tea break.
Nicolls tried to play a sweep to Dananjaya bowling around the wicket and was given out lbw. He reviewed but television replays showed the ball was hitting leg stump.
Nicolls made 42 from 78 deliveries.
B.J. Watling (1) tried to pull a shorter delivery from Dananjaya but was too quick on the shot and was also trapped lbw right before tea.
Taylor was 70 not out from 106 balls with four boundaries, his 31st test half-century, and holding New Zealand's innings together on the opening day of the series.
Dananjaya struck twice to remove opener Tom Lathan and skipper Kane Williamson for a duck in the same over in the first session to reduce New Zealand to 71-3.
Latham and Jeet Ravan had combined for a patient opening stand of 64 from 160 deliveries after New Zealand won the toss and batted.
They negotiated seam bowlers Suranga Lakmal and Lahiru Kumara well and also, for the most part, the three Sri Lankan spinners who were assisted by early bounce from the pitch.
Latham, who scored 30 from 88 balls, went back foot to a bouncing delivery from Dananjaya in the 27th over and edged to recalled wicketkeeper Niroshan Dickwella.
Williamson, the prize wicket, was out three balls later without scoring, also beaten by the bounce when he pushed the ball straight to Sri Lanka captain Dimuth Karunaratne at short midwicket.
Ravan was on 33 right before the interval when he played forward defensively and edged to Dhananjaya de Silva at slip.
This is the first match for both teams in the World Test Championship, and New Zealand also picked three spin bowlers expecting the Galle wicket to turn.