Gus Atkinson claimed a hat-trick before England's batsmen let loose to power the visitors 533 runs ahead of New Zealand on Saturday and in full control of the second Test.
The home side will need to chase an enormous score in the fourth innings after England went to stumps at 378-5 in Wellington on day two.
Captain Ben Stokes resisted any urge to declare, instead cementing England's position of power as four batters posted half-centuries to build on their first-innings advantage of 155.
Joe Root was at the crease on 73 while Stokes struck a whirlwind 35 not out against a tiring attack, after Jacob Bethell and Ben Duckett were both dismissed in the 90s.
The fast-moving nature of the Test means the Black Caps should have ample time to reach any target, as they bid to square the three-match series after losing the opener in Christchurch by eight wickets.
However, history is firmly against them. The highest successful fourth-innings chase at the Basin Reserve is 274, achieved by Pakistan against the hosts in 2003.
New Zealand's hopes of getting back into the Test were scuppered in the opening 40 minutes of play when they lost their last five wickets to be all out 125.
Atkinson (4-31) removed the last three with successive deliveries to become the first Englishman to claim a Test hat-trick since Moeen Ali against South Africa seven years ago.
The seamer was all smiles after bowling Nathan Smith for 14, then having Matt Henry caught in the gully and trapping Tim Southee lbw.
- Bethell crestfallen -
England made batting look a lot easier, most notably when Bethell (96) and Duckett (92) combined for an untroubled 187-run second-wicket stand.
Bethell fell agonisingly short of a maiden Test ton when edging Southee to wicketkeeper Tom Blundell.
The 21-year-old looked crestfallen as he exited after a 118-ball knock which featured 10 fours and three sixes.
Opener Duckett was closing on his fifth Test ton when he played-on off Southee (2-72), ending an innings of 112 balls.
First-Test centurion Harry Brook reached 55 before being caught in the deep off spinner Glenn Phillips while Root put some modest recent form behind him to post another batting milestone.
Root became only the fourth player to reach 50 runs in 100 different Test innings.
Ollie Pope fell for 10 off seamer Matt Henry (2-76), who earlier dismissed Zak Crawley for eight, continuing a lean series for the opener.
In 19 Test innings against the Black Caps, Crawley has scored just 193 runs at an average of 10.15.
In-form England seamer Brydon Carse shapes as a key figure for the remainder of the Test, finishing with 4-46 in New Zealand's first innings, including both overnight batters after they had resumed at 86-5.
Blundell was bowled for 16 before nightwatchman Will O'Rourke was out lbw for a 26-ball duck.
New Zealand added 39 runs off 8.5 overs in the morning, with Phillips not out on 16.