Anderson had Martin Guptill (0) caught by Ian Bell at slip in the third over of the match ??? which only began at 1.30pm due to inclement weather ??? to reach the milestone in his 104th Test, 12 years after his debut.
The England seamer then pocketed Kane Williamson (0) in the same over as the Black Caps slipped to 2-2 after being put into bat by Alastair Cook.
But fifties from Test debutant Luke Ronchi (88) and Tom Latham (84) and a swashbuckling cameo by Brendon McCullum (41 from 28 balls) helped the tourists fight back against their unchanged hosts.
Anderson???s dismissal of Guptill made him just the 12th man to claim 400 Test wickets and only the second current player, alongside India spinner Harbhajan Singh (413 wickets), to do so.
The teams were ordered off the field immediately after Anderson’s strike with the rain returning but it wasn???t long before they were back and Williamson ??? a first-innings centurion at Lord???s ??? feathered behind to Jos Buttler.
Latham, though, played some beautiful shots either side of the wicket en route to his fifth Test fifty, sharing partnerships of 66 with Ross Taylor (20) and 55 with the uber-aggressive McCullum before tea.
McCullum chipped the first ball after the interval, bowled by England???s Lord???s hero Ben Stokes, straight down Mark Wood???s throat at mid-off and when Wood castled BJ Watling (14), New Zealand were five down with 144 on the board.
However, Ronchi, into the side for back-injury victim Corey Anderson, overcame a nervy start to his innings to unleash some powerful strokes and register the quickest Test half-century at Headingley, from just 37 balls.
Latham looked fidgety as he approached three figures and after being dropped three times ??? Wood, Gary Ballance and Cook the guilty parties ??? his luck ran out when he edged Broad to Joe Root to end a 120-run stand with Ronchi.
Ronchi, who struck 13 fours and three sixes in his thrilling knock, followed his team-mate back into the hutch less than two overs later, becoming Broad???s third victim of the innings after clothing a short ball??to Anderson at fine leg.
Wood then had Tim Southee (1) pouched on the deep-square leg boundary by local boy Adam Lyth, before Mark Craig (16) and Matt Henry (14) guided the visitors ??? if not convincingly ??? through to stumps.