Cook is determined to continue his Test career under new skipper Joe Root and the left-handed opener, England's record Test run-scorer, was in excellent form at Taunton.
He eventually fell for 110 when he pulled a short ball from South African batsman Dean Elgar, bowling his occasional left-arm spin, to James Hildreth at mid-wicket.
England Test captain Alastair Cook steps down
Cook batted for over four hours, facing 214 balls and hitting 16 fours. The match was a personal triumph for Cook, who also made 52 in Essex's first innings 129.
His exit on Sunday left Essex 216 for two but by then the visitors were in sight of a victory target of 255, which they achieved without further loss.
Cook resigned as England's Test skipper following their 4-0 series loss in India late last year.
Root says he’s ready to captain England in Tests
Now that he no longer plays international white-ball cricket, the 32-year-old Cook is not due to represent England again until the first Test against South Africa at Lord's on July 6.
While Cook had a hundred to celebrate, it was a different story at Southampton where the Hampshire duo of Michael Carberry and Rilee Rossouw both fell just short against county champions Middlesex.
Former England batsman Carberry, in his first home championship match since being diagnosed with cancer last year, resumed on his overnight 84 not out, but on 98, was caught behind off Tim Murtagh.
South Africa's Rossouw got to within one run of a hundred while batting with a damaged left hand before he was run out by Australia-born ex-England batsman Sam Robson going for a non-existent second run.
Fast bowler Kyle Abbott, who like Rossouw has cut short his South Africa career to play for Hampshire, struck a maiden Championship fifty as the hosts made 438 in reply to Middlesex's 356.
Abbott then removed Middlesex's top three and at stumps they were 111 for four — a lead of just 29 runs.
Meanwhile there was success for two spinners in south London as Lancashire slow left-armers Simon Kerrigan and Stephen Parry shared five first-innings wickets as Surrey were made to follow-on at The Oval.
The days when every county could be relied upon to field two specialist spinners, as Surrey did when the England duo of Jim Laker and Tony Lock ran through opposition batting line-ups on turning pitches at The Oval when the county won seven successive Championship titles in the 1950s have long gone.
Early season English conditions can often be unhelpful to modern spinners but Kerrigan and Parry played their part as Surrey were dismissed for 319 in reply to Lancashire's 470.
At stumps, Surrey were 55 for one in their second innings, still 96 runs behind.
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