Left-armer Martin, the only specialist spinner in the side after New Zealand left out Doug Bracewell from what would otherwise have been an all-seam attack, struck with his seventh ball to remove Nick Compton for 16.
And 43 for one should have become 48 for two when Jonathan Trott, then on nought, miscued a drive only for Martin to drop the caught and bowled chance as he went to his right.
Left-hander Cook, who won the toss, and Compton had made largely untroubled progress against the new ball, with the sunny blue skies overhead making conditions ideal for batting.
But New Zealand applied pressure with four consecutive maidens, including one from Martin, in the lead-up to Compton's exit.
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Martin then floated a delivery up to Compton and the batsman, trying to hit 'inside out', miscued and was well caught off the edge by Tim Southee, running back at point.
But at lunch he still had fine figures of one wicket for eight runs in five overs, including three maidens.
England's fifty took 154 balls and featured just two fours, both struck by Cook.
Earlier, Graeme Swann, fit following an elbow injury, replaced Monty Panesar in the only change to the England side that drew the third Test against New Zealand at Auckland in March, when wicket-keeper Matt Prior's unbeaten hundred helped stave off defeat and ensure a three-match series ended all square at 0-0.