When rain led to an early tea on the fourth day, England in their second innings were 261 for four, a lead of 127 runs.
Cook was 105 not out and Ben Stokes unbeaten on 22.
Together with Joe Root (84), Cook put on 158 for the fourth wicket before the Yorkshireman, for the second time this match, fell within sight of a century.
This was left-handed opener Cook's second hundred in as many Tests after he ended a near two-year-wait for a Test ton with 105 against the West Indies in Barbados earlier this month.
New Zealand had a first-innings lead of 134, with Kane Williamson making an impressive 132.
England resumed on 74 for two, 60 runs adrift, with Cook (32 not out) and Bell (29 not out).
They had not added to their score when, off just the third ball of Sunday's play, Bell, trying to drive a Tim Southee outswinger, edged high through to Tom Latham, deputising for injured wicket-keeper BJ Watling.
In the face of testing pace bowling, Cook responded with a 122-ball fifty that took the left-handed opener three hours.
At lunch, England were 175 for three, a lead of 41, with Cook 69 not out and Root unbeaten on 58.
As the ball became older, things got tougher for New Zealand's quicks while off-spinner Mark Craig, given the difficult job of succeeding the retired Daniel Vettori as New Zealand's specialist slow bowler, was 'milked' for 54 runs in 10 overs with both Cook and vice-captain Root sweeping him to good effect.
Root, who made 98 in the first innings of this match, faced 139 balls including 12 fours.
England were now 232 for four, effectively 98 for four, with the first of this two-match series still in the balance.
Despite New Zealand captain Brendon McCullum setting an unorthodox 8-1 offside field, a composed Cook off-drove Henry for three to complete a 206-ball century featuring 12 fours in five hours and 20 minutes at the crease.
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