Anderson ripped thro' the clueless Kiwi top-order while skipper Paul Collingwood (3/23) mopped up the tail in a pulsating final session that saw the tourists trample hosts' line-up, plucking seven wickets for 96 runs despite Daniel Vettori's quickfire 50*.
Ross Taylor top-scored for Kiwis with a patient 53 while Vettori left stranded at 42-ball 50 for want of partners.
England added 51 to their overnight first-innings total of 291/5 earlier in the day as Tim Ambrose celebrated his maiden Test century. Ambrose playing his second Test match hit 16 boundaries and two sixes in his 149-ball knock. Mark Gillespie dismissed the English tail cheaply, claiming three of the five wickets while Oram and Mills took one each.
The hosts started off on a poor note as they lost opener Matt Bell in the second over of the innings when a full length delivery off Anderson nipped away late to castle the in-form right-hander. Anderson then sent Jamie How packing caught Strauss at first slip to reduce hosts to 9/2 in the sixth over.
The Lancashire seamer, brought back into the team at the expense of Hoggard, removed Matt Sinclair just minutes after the second session. Sinclair paid the price for a lack of foot movement edging a full length delivery for a regulation caught behind.
Fleming (34), in what could be his final innings, along with Taylor staged a recovery act with a fine 71-run stand. The duo capitalised on the early let-offs and ensured no further damage in the second session taking NZ to 100/3 at tea.
England emerged after tea to reassert their dominance, with Anderson again acting as the catalyst.
He induced a half-hearted cut from Fleming which looped to Pietersen at point, and found Taylor"s outside edge as he played back to a ball that left him off the seam to leave the hosts at some sort of bother at 113/5.
Sidebottom then accounted for Oram for a straight forward LBW decision.
The eventful final session saw a flurry of boundaries and wickets. Hosts scored a breezy 97 runs and lost last seven wickets in the process too.
Skipper Vettori joined Brendon McCullum (25 off 21) in the middle and shared a blistering 52-run stand to avert the follow-on threat. The duo looked set for a big one untill McCullum edged Stuart Broad to Strauss at first slip.
Then Collingwood took the baton from Anderson and chopped off the tail claiming the last three wickets.
New Zealand won the first match in the three-Test series in Hamilton by 189 runs.