Monday's nine-wicket success in the second Test at the Riverside gave England an unbeatable 2-0 lead in the three-match campaign.
The fourth and, as it turned out, final day's play also saw the 31-year-old Cook become the youngest player, and first from England, to score 10,000 Test runs during his 47 not out.
That he had a second innings was due to some long-awaited Sri Lankan resistance with the bat.
Their first innings 101, made in response to England's 498 for nine declared featuring Moeen Ali's Test-best 155 not out, followed equally meagre totals of 91 and 119 during an innings defeat in the first Test at Headingley.
"The 10,000 has been a milestone that has driven me over the last few years," said left-handed opener Cook.
"You get tested at the top of the order in all conditions against the best bowlers bowling with the new ball ... And I'm glad I have hung around long enough not to get dropped," added Cook, a veteran of 128 Tests.
As an opener, Cook knows all about the need for mental resilience and he was proud to see England show similar traits.
"They (Sri Lanka) made us work hard for it, and sometimes it is a bit sweeter that way because you feel you have earned it. It was a slog today (Monday) and yesterday (Sunday), and James Anderson's five for 50 in those conditions showed his class," said Cook after the Lancashire paceman was named man-of-the-match following a haul that included his 450th Test wicket.
"He should treasure it - and to be the youngest to 10,000 is incredible."
Anderson, England's all-time leading Test wicket-taker, jokingly pointed out how Cook never drinks more than four pints at the bar.
Cook confirmed the limit was still in place, regardless of the fact he'd just made English cricket history.
"Okay, I'll be having my three-and-a-half pints," he said.
"One of them will be a (shandy) top ... As long as Jimmy buys me one.
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