Having survived a tight lbw appeal early on, the former England captain went on to reach three figures off 202 balls with 16 fours.
The Essex opener shared a 134-run second-wicket stand with Tom Westley but fell for 110 as he pulled a short ball from Dean Elgar to James Hildreth.
But, Westley's unbeaten 86 ensured the visitors chased down 255 in 84.1 overs.
In his first competitive match since stepping down as skipper of the national side, Cook rarely looked troubled while at the crease.
Resuming overnight unbeaten on four, Cook signalled his intent by hitting two boundaries from the first over of the day before Craig Overton was convinced the England batsman had been trapped lbw only for umpire Graham Lloyd to remain unmoved.
Despite 30 wickets falling in the first two days, the hosts' bowlers struggled and Nick Browne shared an 82-run opening partnership with the England opener before being caught behind off Jamie Overton for 35.
Cook batted on unfazed, as new Somerset captain Tom Abell ran out of options, and left the visitors requiring just 39 runs to win when he was dismissed.
A rejuvenated Westley, who made just 10 in the first innings, saw Essex home as he drove seamer Lewis Gregory to the boundary.
Alastair Cook's masterful century guided Essex to an eight-wicket victory over Somerset at Taunton.