Conway puts New Zealand on top after rainy day one against Sri Lanka
The stolid opener's 78 from 108 balls on a grassy pitch gave the hosts a flying start after Sri Lanka captain Dimuth Karunaratne won the toss and elected to field at the Basin Reserve.
In-form number three Kane Williamson was 26 not out, with Henry Nicholls on 18, the pair having built a 37-run stand for the third wicket before stumps was called about 20 minutes early.
After New Zealand took a 1-0 series lead in a final-ball thriller in Christchurch, the second test started more with a whimper than a bang as rain washed out the entire first session.
Only 48 overs were bowled and gale-force winds hammered the ground throughout, blowing off the bails and causing the umpires to chase down their hats across the turf.
Karunaratne had hoped his seamers could grab early wickets but the wind proved as big a test as New Zealand's openers, who put on an 87-run partnership before Tom Latham pulled Kasun Rajitha straight to Prabath Jayasuriya to be caught for 21.
Conway sailed past his half-century but was caught and bowled by spin-bowling all-rounder Dhananjaya de Silva.
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"I was gutted, absolutely gutted when I got out when I did," said Conway.
"I put in all the hard work ... Those are sort of the ones you want to kick on and make the most of."
Sri Lanka went with three rather than four seam bowlers but had little assistance from the wicket.
"Certainly today was probably the windiest I've played at the Basin," added Conway.
"It certainly was a bit of a challenge for them, we took note of that as batters and tried to put them under as much pressure as we could."
Sri Lanka dropped wicketkeeper Niroshan Dickwella after he failed with the bat in Christchurch and dropped Williamson who went on to score a match-winning century.
Wicketkeeper-batsman Nishan Madushka made his test debut in Dickwella's place.
Pace bowler Doug Bracewell came in for the injured Neil Wagner in New Zealand's 11, joining his cousin and all-rounder Michael Bracewell in the side.
Sri Lanka are hoping to win their first test in New Zealand since 2006.