Latham and Conway put on an unbroken 201 for the second wicket after Latham and Will Young, who made 54, produced a 148-run opening stand.
Latham had been working on his timing since a disappointing first Test, where he scored one and 14, and batting coach Luke Ronchi said the progress was obvious.
“It was amazing,” Ronchi said.
I don’t think it’s done as much as expected but at the same time I don’t want to make any excuses for us. I don’t think we bowled as well as we can
“There’s obviously going to be times on a green surface when you’re going to get a bit of luck but you’ve also to keep batting and make the most of it. Some of his (Latham’s) drives and the sounds off his bat were awesome.”
Hagley Oval requires bowlers to maintain a good line and length to deliver wickets on its verdant surface, and Bangladesh bowling coach Ottis Gibson said they were guilty of straying too often.
“I don’t think it’s done as much as expected but at the same time I don’t want to make any excuses for us. I don’t think we bowled as well as we can,” Gibson said.
“We didn’t bowl enough good balls to create the sort of pressure that we created last week. When we got the length to full there was not a lot of seam movement and the New Zealand batters were able to capitalise.”
Taskin Ahmed, Shoriful Islam and Ebadot Hossain fed New Zealand a diet of loose deliveries and as a result Bangladesh spent the day chasing the ball to the boundary at a ground where the average first innings score is 263.
Latham’s unbeaten 186 includes 28 fours, while Conway has 10 fours and a six in his 99.
The hundred moved Latham up to fourth on the list of New Zealand century-makers headed by Kane Williamson with 24.
The only time he was troubled was in Ebadot’s opening over when he was twice given out lbw only for the dismissals to be overturned on review.
Bangladesh appealed vociferously late in the day for Latham to be caught behind, but replays showed the ball missed the bat.
After toiling fruitlessly through the morning session, Bangladesh missed a chance for a wicket in the first over after lunch in what turned out to be a seven-run farce.
A Will Young edge off Ebadot was dropped by Liton Das at third slip allowing New Zealand to run for three.
As the bowler and fielders expressed their frustration, the return throw went to the bowler’s end where no one was backing up and the ball raced through to the boundary.
Much to Ebadot’s annoyance, Young scored a five in his next over when he took a quick single and Mehidy Hasan’s throw at the stumps from mid-wicket raced to the fence again.
Young was to make a further 21 runs before an attempted drive was grasped by Mohammad Naim at point to give Shoriful Islam the first and only wicket.