At the close of play on Friday, England were 159 runs behind with Graham Thorpe on 30 and nightwatchman Matthew Hoggard, who survived an excellent lbw appeal, on nought.
Andrew Flintoff scored an assured 54 before a quicker ball from Cairns struck him in front of middle stump.
Marcus Trescothick made a fluent 63 before mistiming a drive to Styris at slip off the bowling of the impressive James Franklin. Michael Vaughan scored 61 off 65 balls, looking far more comfortable batting at four than he did in the second Test.
But former Nottinghamshire all rounder Cairns, who has chosen to retire at his old home ground after 62 Tests, trapped Vaughan lbw with a slower delivery that kept low.
The captain and his deputy had steadied the innings after Andrew Strauss was out for a duck edging Cairns to Brendon McCullum and Mark Butcher, who fractured a finger in the field, glanced Franklin to Styris at slip for five.
New Zealand's bowling options, already hampered by injuries, were further reduced when debutant Kyle Mills (side strain) and Chris Martin (hamstring) both left the field.
COMPOSED CENTURY
Earlier a composed century from Styris had helped the tourists to 374 for seven at lunch but England quickly wrapped up the innings to leave the Kiwis short of the large total possible on a relatively flat Nottingham pitch.
Styris continued where he left off on Thursday evening playing with confidence despite a poor tour with the bat so far.
He was finally out for 108 when he came down the pitch to Ashley Giles and hit a leading edge to substitute fielder Bilal Shafayat.
Hoggard then captured his 100th Test wicket when Geraint Jones snaffled an edge off Mills for nought, though television replays showed the ball touched the ground just before the wicketkeeper scooped up the ball.
The Yorkshire swing bowler's next wicket was clean however, with Vaughan taking a juggling catch to dismiss Martin for two.
Hoggard was also in the action when he managed to cling on at the third attempt to a catch at third man from a lofted McCullum cut for Steve Harmison's third wicket.
Seamer Martin Saggers had earlier forced Jacob Oram (14) and Cairns (12) to sky the ball into the hands of Strauss and Thorpe respectively.
England lead the three-match series 2-0 and are looking for their first series whitewash over New Zealand since 1978.
