Conway fell short of a hundred as he was caught and bowled by Dhananjaya de Silva, who dived low to his right to get both hands on a superb catch, to end a thrilling 108-ball knock.
First-Test hero Kane Williamson was unbeaten on 26 at the close of play alongside Henry Nicholls on 18 when bad light stopped play at the Basin Reserve after Sri Lanka won the toss and put New Zealand in to bat in blustery conditions.
New Zealand started strong, with Conway and Tom Latham putting on an opening partnership of 87, after the 2,500th Test match in cricket history was hit by a damp start.
Heavy overnight rain delayed the toss until after lunch.
Latham fell for 21 around the halfway point of the day’s curtailed proceedings when he spooned a Kasun Rajitha delivery to deep square, into the hands of Prabath Jayasuriya.
Conway responded by smashing 13 fours in a fiery knock.
SCOREBOARDNew Zealand 1st innings
Latham c Jayasuriya b Rajitha 21
D. Conway c & b de Silva 78
K. Williamson not out 26
H. Nicholls not out 18
Extras (lb10, nb1, w1) 12
Total (2 wickets; 48 overs) 155
Fall of wickets: 1-87 (Latham), 2-118 (Conway)
Bowling: K. Rajitha 16-3-42-1, A. Fernando 12-4-33-0, L. Kumara 10-1-49-0, D. de Silva 8-3-18-1, P. Jayasuriya 2-0-3-