The duo eclipsed the mark of 276 set by Jackie Mills and Stewie Dempster in 1930 with an opening stand of 317 as England were made to suffer in the sunshine.
The home side fought back well with two wickets in the last two overs of the day, but New Zealand still closed day one of the decider in the series on a commanding 361-4 after Latham and Conway had exploited the batter-friendly conditions to the full.
"It was a great day for us, and it was nice to build a partnership with Tom," Conway told Sky Sports.
"We’re pretty happy with the way things have gone. We knew that the wicket was going to be quite nice. It was important for us to put pressure on the bowling.
"I don't know quite what par is, but it was a bit disappointing to lose two wickets at the end. We've still got plenty of batters in the shed, hopefully we can kick on."
It made for a punishing return for Ben Stokes and Gus Atkinson, who were back after being made unavailable for the second Test at The Oval following an incident at a London nightclub.
An England and Wales Cricket Board investigation found that while both players had broken a team curfew, they were not involved in the violent conduct which saw a Saracens rugby player strike a member of the England team’s security staff.
It was Stokes, the England captain, who finally made the breakthrough in the 73rd over as Latham edged to wicketkeeper Jamie Smith to depart for 151. It made partial amends for a poor drop from Smith, who was back after paternity leave, when Latham gloved Atkinson down the leg side on 129.
Conway followed in the next over, caught at wide long-on by substitute Matt Fisher off Joe Root for 157.
Latham had completed his 17th Test century first just before tea, clipping Stokes away down to the fine leg boundary.
Conway, who made a double century on his Test debut against England five years ago, followed suit shortly after the interval and then accelerated, hitting Shoaib Bashir and Josh Tongue for sixes in successive overs.
He had been given a life on 71 when the off-spinner struck him on the pads with a delivery that would have gone on to the stumps, but England failed to refer the decision to the TV umpire.
Bashir, brought on after just 47 minutes, bowled 22 overs on day one in the sweltering conditions, taking 0-97.
England’s bowlers kept toiling away, and Atkinson had his reward in the penultimate over when Rachin Ravindra miscued a pull and skied a catch to Smith before Jofra Archer, who had struck Henry Nicholls on the helmet earlier, found his outside edge with the last ball of the day.
"Test cricket is hard, and on a surface like that it’s particularly hard. So I think the effort was pretty good throughout the day," said England fast bowling coach Tim Southee.
"We got some rewards late in the day, which should give the guys some energy going into tomorrow with a new ball in hand."
Nevertheless, it was a strong start to the Test for a depleted New Zealand, who were without Glenn Phillips and Matt Henry because of injury and Kyle Jamieson, who is being rested.