England were cruising on the second day of the second Test while left-handed openers Lyth (107) and skipper Cook (75) were sharing a first-wicket stand of 177.
But the departure of Lyth, needlessly run out shortly before the advent of the new ball, sparked a slump that saw four wickets fall late in the day.
At stumps, England were 253 for five in reply to New Zealand's first innings 350, a deficit of 97 runs.
Lyth, 27, was Cook's sixth opening partner since the retirement of former captain Andrew Strauss, with Nick Compton, Joe Root, Michael Carberry, Sam Robson and Jonathan Trott all given a chance in the problem position.
However, Lyth went some way to making the berth his own ahead of the Ashes series starting in July.
Today saw England's first century opening stand in a home Test since Cook and Strauss added 186 against India at Edgbaston on 2011.
But he was far more assured back in familiar surroundings, although he did have several 'nervous nineties' moments.
On 90, he almost played on to Southee, kicking fresh air, as he tried to boot the ball away from danger. In fact, the ball hit the stumps but didn't dislodge either bail.
Lyth was on 94 when his drive against off-spinner Mark Craig appeared destined for mid-on.
But substitute fielder Neil Wagner went the wrong way as the ball flew past him for a four that took Lyth to 98.
Earlier, Cook overtook Gooch's previous England record of 8,900 Test runs when he square-drove Southee for four that gave him the score of 32 he needed to surpass his mentor's mark.
The economical Craig removed Cook, although it needed a New Zealand review to have the skipper lbw after he missed an intended sweep.
Cook batted for nearly four hours and faced 187 balls including a dozen boundaries.
He walked off having moved up to 12th place in the all-time list of leading Test run-scorers with 8,944 runs in 114 matches and 203 innings, at an average of 46.82, with 27 hundreds and 41 fifties.
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