Essex 295 for 7 (Cook 109, ten Doeschate 102, Taylor 3-65) beat Sussex 285 (Nash 66, van Zyl 61) by 10 runs
From the depths of 19 for 3, Cook and ten Doeschate ensured Essex posted a competitive 295 off their 50 overs - and that proved too much for Sussex, who finished 11 runs short.
Cook took his total to 317 runs in five one-day innings with a 131-ball 109, his second three-figure score of the season. He shared a fifth-wicket stand of 142 with ten Doeschate, who was run out off the last ball for 102 for 91 overs.
Chris Nash and Stiaan van Zyl both scored sixties in Sussex's reply, but no one went on to make the big scores that Cook and ten Doeschate had earlier.
Sussex failed to capitalise on a great start when West Indies international Taylor struck with the first three balls of his third over to underline why both captains would have put the other team in.
Varun Chopra was the first Essex batsman to go, chasing a wide one to give Michael Burgess a diving catch behind for two.
Tom Westley followed, attempting to withdraw his bat but only managing to offer a faint touch to the same combination for his second golden duck in successive innings.
Adam Wheater was the hat-trick victim when trapped plumb lbw on the back-foot playing down the wrong line.
Taylor's fourth ball of the over, however, was eased through midwicket by Dan Lawrence for four to offer some inspiration to the shellshocked Essex batsman. When Taylor was rested after his opening six-over salvo he had figures of 3 for 23. By the end he had been on the receiving end of a battering as his 10 overs went for 65 runs.
The fourth-wicket pair of Cook and Lawrence set painstakingly about repairing the damage. They put on 65 runs in 15 overs before Lawrence pushed a return catch to spinner Danny Briggs's fifth ball for 34.
When he was 38, ten Doeschate survived what looked a comfortable caught and bowled chance. He hit the ball almost vertically in the air, so high into the stratosphere that Shahzad was able to race down to the striker's end before dropping the plummeting ball right by the stumps.
Luke Wright, Sussex's captain, did not hide his displeasure. "It came down to that moment when it's never a bowler's catch," he said. "It goes up and it's straight to a keeper with big gloves on and then he would take the catch.
"I think everyone in the ground was shouting, 'keepers!' But, look, the bowler wanted it and unfortunately he dropped. At that point it's a big moment and Tendo made use of it."
Sussex were made to pay as he accelerated with a succession of bludgeoning boundaries mixed with tip-and-run singles turned into twos.
Ten Doeschate had just passed fifty - from 57 balls - when Cook swept a hip-high delivery from Shahzad for his 12th four to reach a 122-ball century. He was finally out when he cut Jofra Archer backward of square to Shahzad.
Ashar Zaidi had hit Wiese for a straight six before he went for another and was held above his head by Briggs on the square-leg boundary. Ten Doeschate swatted a short delivery from Archer to reach three-figures from the penultimate ball of the innings and was run out from the last. Essex had added 99 in the last 10 overs.
Nash and Wright played sensibly as they laid the foundations for the chase with a first-wicket stand of 67 in 14 overs. But the ball after Wright had driven Jamie Porter for a flat, straight six, he hooked straight to Paul Walter at deep fine leg for 32.
Harry Finch lasted four balls before he went lbw to one from Zaidi that straightened. After which Sussex became becalmed for a spell, and Nash and Stiaan van Zyl went 36 deliveries between boundaries, during which time they managed just nine singles.
Nash went for a 90-ball 66 when he played a cross-batted shot to Matt Quinn and was caught at mid-on. But van Zyl passed fifty from 60 balls with a lofted drive to long leg for four, followed by a six over long-off, before he was caught on the ropes by Walter for 61. Half the side were out with 108 still required from 10 overs when Burgess was bowled by Simon Harmer. Laurie Evans perished to Cook at wide mid-on for 30 and Wiese fell to another attempted big-hit at long-off by which time the asking rate had passed 12 an over.
Sussex gained a second wind when Quinn's nine-ball over went for 21 runs. Walter restored order with the first ball of the next over when he bowled Briggs. But Archer smashed his second six before Walter ended the 21-ball slog-fest by having the Barbadian caught on the square-leg boundary for 45 and then claimed his third wicket by bowling Taylor.