Guptill was caught by Alastair Cook at first slip off Mark Wood just before the interval but the pace bowler was denied his first test wicket because replays showed he had over-stepped the crease and the delivery was called a no ball.
Guptill (26 not out) and Latham (13) were otherwise largely untroubled by the England pace attack in overcast conditions at the home of cricket.
New Zealand took three wickets in the first hour to wrap up the England innings after the hosts resumed on 354 for seven.
Moeen Ali reached his fifty before edging left-arm seamer Trent Boult to stand-in wicketkeeper Latham for 58.
Stuart Broad's miserable batting form continued when he nicked Boult through to Latham for three and James Anderson, on 11, was caught and bowled by Matt Henry.
Boult completed figures of four for 79 and Henry took four for 93 on his debut.
Joe Root made 98, Ben Stokes 92 and Jos Buttler 67 on Thursday to help England recover from a dismal start in which they slumped to 30 for four. – Reuters
Original source: Black Caps off to a steady start