Essex 107 for 2 trail Kent 207 (Blake 89) by 100 runs
The England captain had reached 49 off 93 balls when a yorker by Rabada off the fifth ball of the last over trapped him lbw
It was a disappointing end to a day in which Essex had been in full control of the top of Division Two clash from the start.
It had not been plain sailing before that, either. Rabada had Cook nervously dancing around in his crease at the start in the face of some serious pace and the awkward bounce that had helped the Essex bowlers so much earlier in the day.
This was Rabada's first red-ball outing since he took 13 wickets against England at Centurion in the fourth Test. The England captain was not one of his victims then, but he had been out twice to him in the second Test.
Cook and Nick Browne saw off Rabada's fierce opening spell, even if they had only reached 20 off 11 overs. Rabada rested after six overs, which had gone for just nine runs, before returning near the close after a 10-over break.
Cook freed up with Rabada out of the attack, caressing the first four of his innings through the covers off Mitch Claydon. He repeated the punishment a couple of overs later as Claydon went for three fours through the off-side, one square, another through 45 degrees and the third through the covers.
Browne was the first to go, getting to the ball late and spooning Rabada to Sam Northeast at mid-on for an 84-ball 33.
Essex had taken advantage of a contested toss and put Kent in on a green wicket, and one on which there was swing through most of the first two sessions and some erratic bounce.
Only Alex Blake, with an unbeaten 89, his highest Championship score of the season, showed much resistance to the Essex seamers. Led by Jamie Porter's 3 for 51, Essex had reduced Kent to 48 for 4 when Blake arrived at the wicket mid-morning.
His stay of 116 balls included 13 fours and he dominated a seventh-wicket stand of 56 with Callum Jackson that helped Kent post a larger total than looked likely at one stage.
Kent's wickets fell evenly between the morning and afternoon sessions with the visitors going to lunch at 88 for 5.
Fabian Cowdrey was first to go when Graham Napier sent his leg-stump cartwheeling with an inswinger. Joe Denly and Sean Dickson survived some close calls before they departed within four balls, both to catches behind by James Foster off Ravi Bopara and Porter respectively.
Kent captain Northeast became Foster's third victim soon after when he wafted outside off stump to Matt Quinn and Darren Stevens was trapped lbw by Bopara.
Adam Ball went soon after lunch, but not before he had been dropped at third slip by Nick Browne and had a visit by the physio to treat a hand injury. That was clearly on his mind as he tried to fend off a lifter from Porter and the ball spiralled to Jaik Mickleburgh at gully.
That brought together Blake and Jackson to steady the innings. Blake reached his half-century off 72 balls with his 10th four.
Jackson, making his Championship debut in the absence of Sam Billings, who is on England duty, was happy to play the junior role. He was eventually out for 19 when he tried to push Ryan ten Doeschate through the covers, missed and was trapped lbw.
The last three wickets added another 50 runs to take Kent past the 200 mark. James Tredwell lost his middle-stump to become Porter's third wicket, Rabada was bowled by Dan Lawrence and Claydon dollied one up to Mickleburgh in the gully off Napier.