England, at tea on the second day of the second Test, were 163 without loss in reply to New Zealand's first innings 350 -- a deficit of 187 runs.
Lyth, on his Yorkshire home ground, was 80 not out after he managed just seven and 12 on debut during England's 124-run win in the first of this two-Test series at Lord's.
Meanwhile fellow left-hander Cook was 71 not out, having gone past mentor Graham Gooch's England Test runs record of 8,900 when he made 32.
However, the 27-year-old Lyth went some way to making the berth his own ahead of the Ashes as he helped Cook compile England's first century opening stand in a home Test since Cook and Strauss added 186 against India at Edgbaston on 2011.
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Cook, whose 162 at Lord's last week was key to England's victory, overtook Gooch when he square-drove Tim Southee for four.
A modest wave of the bat followed for, as Cook had made clear in the build-up to this match, it would not benefit his side much were he to break Essex predecessor Gooch's record and get out soon afterwards.
Cook was less convincing in posting his fifty, an edged cut off Henry flying over the slips and down to the boundary to give the 30-year-old a ninth four in 118 balls.
Lyth then showed plenty of poise and touch to ease Trent Boult off the back foot through the covers for four.