Sussex resumed on 355-8 but Jamie Porter bowled Ajmal Shahzad and Steve Magoffin in the first over of the day to wrap up their innings for 360.
The hosts reached 77-3 at lunch as Cook edged Magoffin to second slip and Nick Browne and Ravi Bopara fell cheaply.
Westley (86) and Jesse Ryder (51) boosted Essex as they closed on 252-7.
Cook complied with new helmet regulations while batting during the match, after being spoken to by the England and Wales Cricket Board following Essex's victory Gloucestershire last week.
Tom Westley starred for Essex after England Test captain Alastair Cook made just one as the visitors struggled on day two against Sussex at Hove.