All in all, England managed just 17 runs for their last three wickets.
If that gave England's seamers cause for optimism, the opening exchanges successfully dampened it.
Where Southee and Boult found consistent swing, the new ball simply refused to move for Anderson and Broad.
After 10 overs Peter Fulton and Hamish Rutherford reached 34 for nought, sharing six boundaries.
Anderson was unhappy with the ball and eventually persuaded the umpires to change it.
There was no immediate benefit, with Finn's first two overs costing 13 and Rutherford playing the shot of the morning off Anderson, an imperious back-foot drive through cover.
The New Zealand 50 came up in just 74 balls, a sign of how innocuous England's seamers had been in the period.
When they did break through it was a surprise, Finn finding just enough bounce to take Fulton's leading edge. The ball looped in the air and provided a simple return catch.
The wicket boosted Finn, who started to find some seam movement and soon had Rutherford in trouble.
He beat the outside edge three times in a row then found it at the fourth attempt, Ian Bell snaffling the chance at gully.
England showed good control after tea, Stuart Broad providing the control and Finn hunting more wickets.
It took him less than two overs to succeed, bringing a ball back into Ross Taylor and bowling him off the inside edge.
Swann did not have to wait long to make his mark on the game, though, captain Alastair Cook calling for his services with New Zealand 76 for three after 26 overs.
His impact was devastating and after two overs, he had the tourists rocking at 82 for six.
Swann first reprised his old habit of striking in his first over, drawing Dean Brownlie out with drift and then spinning the ball through the gate to take out off stump.
He repeated the trick at his next visit, the same combination of drift and turn doing for new man Martin Guptill.
Swann was firmly in the zone now and seconds later added the scalp of Kane Williamson with an lbw that was turned down on the field but approved on referral.
That brought the aggressive pair of Brendon McCullum and Southee together and they were quick to play their shots, despite the parlous position.
A run of boundaries, not all entirely authentic, took the total past 100 and the pair had added 37 by the time Broad joined the wicket-takers.
He went full and straight to Southee, who was confirmed lbw for 19 only after DRS showed that the ball hit pad marginally before bat.
The follow-on was now a distinct possibility and Swann brought it even closer when he had Doug Bracewell held by Bell at silly point.
It seemed that New Zealand's last hope expired when McCullum was ninth out for 20, waving the bat at a Broad delivery that demanded more respect.
But Boult and Neil Wagner put on a dashing half-century for the last wicket.
Boult (24 not out) stunned England by launching Swann for three sixes down the ground, while Wagner took 18 from a single Broad over before Anderson bowled him for 27.
Cook opted not to send New Zealand in again, frustrating some fans who sensed blood.
