Warwickshire 161 and 67 for 1 need a further 336 runs to beat Essex 245 and 316 for 9 (Cook 83, Lawrence 74, Rhodes 4-38)
Liam Banks dug in with great determination as Warwickshire set about batting out a minimum of 121 overs to prevent a heavy defeat at Chelmsford.
Banks was 36 not out as Warwickshire reached 67 for 1 in their pursuit of 401 to pull off a mission improbable and record a third victory of the season. More realistically, the capture of another nine wickets stand between Essex and the seventh victory in eight games that would take them to the top of the Specsavers County Championship by just four points, assuming Yorkshire do not let slip their stranglehold over Somerset at Headingley.
The Warwickshire openers took advantage of some wide open spaces in the field to pass fifty in the 13th over. But Aaron Beard's arrival into the attack heralded the breakthrough Essex wanted with Will Rhodes lbw for 25. Warwickshire made it to the close without further alarm, still 334 runs from their target.
The match had essentially been put beyond Warwickshire's reach during an eighth-wicket rampage between Dan Lawrence and Simon Harmer that extended Essex's lead from 299 to nearly 400 in just 20 overs. Both batsmen fell straight after tea, but not before they had put 84 valuable runs on the board. Lawrence departed to the second ball of the final session, charging Rhodes and nicking to Tim Ambrose for 74.
That gave Rhodes his fourth wicket of the innings, and ninth of the match, at a combined personal cost of 55 runs. The part-time seamer's first-innings 5 for 17 was a career-best; his 4 for 38 in the second constituted the next best.
Harmer followed without addition, attempting to hit Jeetan Patel over the top and being caught on the long-leg boundary by Banks.
Alastair Cook had laid the foundations in the morning towards the eventual declaration with a carefree 83 from 168 balls. He was the senior partner in a second-wicket stand of 63 off 26 overs with nightwatchman Matt Quinn that frustrated Warwickshire for an hour and a half. Quinn contributed nine to the partnership from 69 balls.
Cook hit 11 boundaries in his sixth Championship half-century of the season, and went on the attack once he had reached that milestone, hitting Patel for three of them in quick order.
The return of Rhodes accounted for Cook when he was pinned lbw. The Yorkshireman had another in the same over when Tom Westley was caught behind wafting outside off stump. Quinn finally departed next over, nicking behind as Essex collapsed from 135 for 1 to 139 for 4.
Rishi Patel fell to the first ball after lunch when he deflected a ball from pad on to bat to Banks at second slip. Brookes claimed a second wicket in the over when Ryan ten Doeschate lost his middle stump as he attempted to force the pace. Adam Wheater had scored 21 at almost a run-a-ball when he reverse-swept at Patel and Rhodes threw himself from first slip to where second slip would have been to take the catch.
Lawrence rattled along apace and clobbered Brookes off the back-foot for four to bring up his second fifty of the match from 101 balls.
Essex took the field in the evening without wicketkeeper Wheater, who had struggled in Warwickshire's first innings after taking a blow to his thumb. Will Buttleman replaced him behind the stumps.