England's leading Test run-scorer and a veteran of 19 years at first-class level, had never previously reached three figures twice in a match but finally did at Chelsmford.
Any outside prospect of a positive result evaporated on the final morning as Yorkshire - needing to build quickly on their overnight advantage of 22 - lost their last five wickets for 39 and were dismissed for 465.
That left Cook, who had scored 107 first time around, and opening partner Nick Browne under little pressure in their second innings and the pair eased comfortably past the deficit with an unbroken partnership of 167 before the captains agreed on a draw at 4.50pm.
Resuming on 425 for five, fast scoring appeared to be a necessity for the White Rose - but they displayed little attacking intent, garnering only three runs from the first six overs of the day.
Dawid Malan's hopes of converting his 87 into a century disappeared when he clipped his first ball of the morning, a Shane Snater half-volley, into the hands of square leg, while Harry Brook showed none of the fluency that had already brought him a hundred.
Brook eventually fell for 123, caught behind wafting at a short delivery from Sam Cook, who was also unlucky not to claim the wicket of Harry Duke when he struck off stump - yet the bail refused to budge.
Cook had to be content with figures of three for 78, while Snater (three for 62) finished off the innings by having Dom Bess caught behind, leaving Essex with a deficit of 62 and half an hour to negotiate before lunch.
They might have gone in a wicket down, with Jordan Thompson finding the edge of Alastair Cook's bat in his third over but Duke, diving across in front of the slips, was unable to cling onto a difficult catch.
After that, the former England captain looked completely in command and, while Browne's progress was more sedate, he gave only one chance en route to his 50 not out, a stumping opportunity off Adam Lyth that Duke failed to execute.
However, it was Cook who took centre stage and he brought up his 72nd first-class hundred from 178 deliveries, pulling a long hop from Joe Root to the fence shortly before stumps were drawn.