The partnership, which spanned 86.4 overs, is the joint 12th highest opening stand in men’s Test cricket and only the eighth stand of 300 for any wicket for New Zealand.
New Zealand’s record partnership was set by Martin Crowe and Andrew Jones, who added 467 for the third wicket against Sri Lanka in Wellington in 1991.
Conway faced 279 balls and hit 25 fours, compiling his sixth Test century in the process.
Latham, who earlier won the toss, was dropped by wicketkeeper Tevin Imlach on 104 before being caught by a diving Roston Chase at first slip off Kemar Roach.
Latham faced 246 balls, struck 15 fours and a six and now has 15 Test hundreds.
“It’s the best I’ve been part of. It was a good time out there,” said Conway.
“We had a discussion this morning around trying to bat big, bat well and give the bowlers an opportunity to bowl in the fourth innings.
“It was a challenging morning session, particularly for me. I was fighting with myself during the first hour. They bowled really well.
“After lunch, the ball got a little bit softer and the wicket sped up, so it offered up some good scoring opportunities for myself and Tom.”
Conway will resume alongside nightwatchman Jacob Duffy when play resumes on Friday (22:00 GMT, Thursday).