Essex 298 (ten Doeschate 78, Browne 61) and 202 for 8 (A Cook 66) beat Kent 387 (Kuhn 140, Robinson 78) and 112 (Harmer 4-35) by two wickets
Essex still required 31 runs when Cook, who turned 23 on Tuesday, and wicketkeeper Wheater came together following a batting collapse created by Ivan Thomas.
Former England captain Sir Alastair Cook and 21-year-old debutant Feroze Khushi had put on 86 runs for the fifth wicket to tilt the balance towards Essex.
But Thomas, who missed the 2019 season after he needed surgery to reconstruct his anterior cruciate ligament, took four wickets in 16 legal balls to give the hosts the jitters.
Then he ninth-wicket pair hung on in front of an unfittingly empty Cloudfm County Ground, Chelmsford, with Sam Cook scoring the winning runs to send Essex top of the South Group table, claiming 21 points to Kent's seven.
Kent had been on top of the match for two and a half days, having boasted a first-innings lead of 89.
But things started to unravel when they were bowled out for 112, thanks to Simon Harmer's four wickets and fast bowlers Sam Cook and Jamie Porter's contributions on day three.
Their first mission on the fourth day was to get nightwatchman Porter out quickly, which Grant Stewart managed in the 10th over - the usual No.10 striking to Daniel Bell-Drummond, who like Sam Cook was celebrating his birthday, at short cover.
That brought Alastair Cook to the crease, where he set up for the long haul - starting with a patient 37-run partnership with usual opening partner Nick Browne.
Kent then forced another twist in the topsy-turvy contest when Stewart and Hamidullah Qadri shared three wickets in five overs, for only 13 runs.
Qadri found Browne's edge to give him his first wicket for his new team after moving from Derbyshire.
Stewart then returned for his second spell of the day and struck immediately when Essex skipper Westley nicked behind, before Varun Chopra followed suit in his next over.
It would undoubtably have been a nervy time to arrive at the crease for only his second first-class innings, but Khushi shrugged off the butterflies to rebuild with Alastair Cook.
While Cook played his trademarked flicks and jabs, Khushi pulled off a number of flowing shots - the best of which came when he elegantly cut and then pulled Thomas in the 48th over. Cook reached his 181st first-class half-century in 93 balls, brought up with a fine legside flick, before the 50 stand came up with Khushi.
Essex then suffered their second wobble of the day as Thomas gave Kent hope.
Cook may have felt a tad aggrieved to have been given out lbw to a ball angling across him, but Khushi could not quibble his edge behind.
That left Essex still requiring exactly 50, and only 20 more runs had been added before Ryan ten Doeschate was lbw to one which stayed low, with Harmer nicking the next ball behind to Ollie Robinson. Thomas returned figures of 4 for 32.
But Wheater and Sam Cook shared the last 31 runs to get Essex up and running in the inaugural Bob Willis Trophy.