The South African drove at his seventh ball from Neil Williams but Paul Grayson, Essex's stand-in captain, put down a knee-high catch at second slip.
Cullinan, who reached his century off 171 balls, shared a stand of 217 in 56 overs with Gerry Liebenberg, who made 96 to show some timely form before the decisive fifth Test.
It took Liebenberg's aggregate against the counties to 569 for an average of 71, a tour mark brought down to 50 by his disappointing Test form.
Cullinan was trapped leg before by Mark Ilott for 157 as the South African declared on 406 for 5.
The Essex openers Darren Robinson and Ian Flanagan survived four overs before close to reach 13 without loss.
In the County Championship Worcestershire are struggling to save the follow-on against Yorkshire. The visitors declared at 455 for 9 and then made their way through the innings to leave Worcestershire 145 for 7 Paul Hutchison produced a 16-over spell either side of lunch which earned him figures of 3 for 38 and included the prized wicket of Graeme Hick.
Paul Nixon rescued his side against Lancashire at Old Trafford, though not without a little luck. After the first day was rained out Leicestershire were 82 for 5 before the wicketkeeper's unbeaten 42 helped his side to 218 for 7 before bad light halted play. He was dropped by Mike Watkinson and when Wasim Akram's yorker bowled him he was rescued by the cry of no ball from the umpire.Reuse content