Fast bowler Cook grabbed two in two with the new ball before Harmer took two more wickets to claim match figures of 8-178 and Porter rounded off the tail.
Somerset had batted stoically in the hope of batting out 151 overs, or reach 466 to win, but they were eventually bowled out for 269 with two sessions to spare.
Essex only won two Championship matches at home last season, having gone unbeaten at Chelmsford between September 2018 and September 2022.
The day’s play was preceded by a minute’s silence for the victims of the Nottingham attacks, while players from both teams and the umpires wore black armbands and the flags were at half mast.
The two students who died both had links to Essex and Somerset cricket. Barnaby Webber, 19, played for Bishops Hull Cricket Club in Taunton, while Grace O’Malley Kumar, 19, played and captained Essex Women from under-11 to under-15.
Somerset had shown great fight and grit on the third evening to give themselves hope of saving a draw, having been bowled out for 167 with a first-innings deficit of 295.
But Tom Kohler-Cadmore’s dismissal to the final ball of the day swung the mentality metre strongly in Essex’s favour.
Tom Abell and James Rew did bat out the first 137 balls of the morning, putting on 56 with great skill.
The second new ball was coming, though, and would prove to be the twist for the visitors.
Cook needed just two deliveries to extract a thin edge behind to see off Abell for 83 before a wobble-seamed delivery nipped into Kasey Aldridge’s off stump next ball.
Craig Overton navigated the hat-trick ball that teased the outside edge but Rew soon fell to Harmer – who had shared the fresh nut.
Left-handed Rew drove at the off-spinner with the edge brilliantly taken at first slip by a full-stretched Alastair Cook.
Matt Critchley took an equally great catch at leg slip when Overton turned Harmer around the corner.
And Essex’s second victory of the campaign was confirmed with Matt Henry tickling behind and Josh Davey sensationally caught behind by Will Buttleman, both off Porter.
Essex take 22 points, to Somerset’s two, to keep their nose in the Championship race.
Captain Tom Westley said: “It feels like it has been while since we’ve won here. We had a huge amount of success for a number of years winning games.
“That was a world-class performance with bat and ball. It shows how dominant we can be when we bat and bowl well.
“I think Chef [Alastair Cook] has found some form in the last couple of games with that 99 at Trent Bridge, so for him to carry on with a hundred was outstanding.
“Critch has been batting well all year but the hundred had eluded him, so it was really good for him to get that hundred at Chelmsford, plus with the ball he broke up partnerships.
“It is really pleasing as a captain when different people put their hands up to help contribute to wins. We’ve struggled with niggles and injuries – like [Doug] Bracewell going down on the morning of the game – but [Ben] Allison came in and took a crucial wicket in the first innings.
“I think the club is in a really good place and that result kickstarts our season.”
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