The third Test against South Africa lay perfectly poised at lunch on day three, with South Africa 16 without loss in their second innings and setting about building a lead.
But Bayliss issued an animated call to arms at the interval and watched on as Stuart Broad produced another magical spell for his seam bowling scrapbook.
Broad tore apart the Proteas top order in a spell reminiscent of his Ashes heroics at Trent Bridge, finishing with 6-17 as the hosts slumped to 83 all out.
The game – and the series – was settled there and then, with England chasing down a target of 74 for a seven-wicket victory and an unassailable 2-0 advantage.
Broad, at his mercurial best, was the obvious focal point but Cook praised the usually taciturn Bayliss for his fiery address.
“Trev gave us a kick up the a*** at lunch,” he said. “Rather than sulking about it the lads looked at themselves, led by Broady, and I think there was a real intensity in the field for that two-hour session. It was a realisation from a guy that doesn’t say too much that this is the time: if you want to win the series then you’ve got an opportunity.
“It seemed to work so well done Trevor. It really helped.”
Broad left the field with an impressive haul – his best overseas figures, a man-of-the-match award, a first series win over South Africa and third place on England’s all-time wickets list overtaking Bob Willis.
His 15th five-wicket haul was another fine example of his ability to alter the course of a Test match when the mood takes him, and he felt he owed the team a display having struggled with a virus on day one. “I was frustrated from the first innings, I was a little bit ill and I didn’t bowl very well,” he admitted.
“But a bit of experience tells you now is the time to step up and make something happen. I felt in good rhythm today, for sure.
“That was the sort of wicket as a seam bowler you’d like to take around everywhere. It offered good movement in the air and off the deck and good bounce.
“When you get the opportunity to bowl on a wicket like that and feel in a good rhythm you’ve really got to take your chance. The most important wicket was actually the first one, making the breakthrough with the newer, harder ball. That gave the team confidence.
“Any time you win a Test match anywhere it is special but to beat South Africa for the first time in a Test series and to do it away from home is very special.”