The 30-year-old left-handed opener started England's reply to New Zealand's first innings 350 today needing just 32 more runs to surpass Gooch's mark of 8,900 runs in 118 Tests.
Cook, first coached by ex-England captain Gooch as a schoolboy, reached lunch unbeaten on 27.
Then, to his 13th ball after lunch Cook passed Gooch -- whose runs came in four more Tests (118 to 114) -- when he square-drove Tim Southee for four to break a record that had stood for nearly 22 years.
"It would be a lovely moment. I wouldn't put myself anywhere near in his class but it would be a great moment," said Cook.
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"If I'm lucky enough to get that it will be a moment I'll remember but if I am 30 not out it's important in a game situation I go on and get a big score," added the left-handed Cook, like the right-handed Gooch an Essex and England opening batsman.
Cook added: "I wouldn't be here, or anywhere near here, without Goochie's help, so it will be slightly strange if I do go past his record because without his hard work and dedication to me and my game, I wouldn't have scored half the runs I would have done.
By going past Gooch, Cook moved up to 13th place on the all-time list of leading run-scorers in Test cricket, where retired India great Sachin Tendulkar remains way out in front with 15,921 runs in 200 matches.