Bayliss praise for 'fantastic' captain Cook
Cook has generally been considered a fine batsman but a limited captain, in the "lead by example" mould, but in Cardiff he shuffled his attack and positioned his field masterfully. Whether it was the placement of two short midwickets, the decision to recall Moeen Ali for an over before lunch on day four or the gamble to start with Stuart Broad on day three with the new ball just 10 overs away, almost everything Cook tried came off.
Even Shane Warne, a vocal and remarkably repetitive critic of Cook's leadership, was moved to admit it might have been his best game as captain.
Bayliss was clearly impressed, too. While he expressed some gentle reservations about Cook's batting in the game - quietly suggesting that he may be well served reverting to the style that brought him 9000 Test runs - he was appreciative of the captain's efforts to set the tone for the team with a somewhat uncharacteristically aggressive innings.
"His captaincy in this game was fantastic," Bayliss said. "He is a very calm operator and he is very well respected by the players in the team. His captaincy was top-notch.
"He is obviously a fantastic batsman, but he didn't score as many runs in this match as he would like. I thought his approach in the first innings - even though he only got 20 - sent a bit of a message to the rest of the players that here was the captain who was trying to be that positive batsman. Maybe that is a little bit out of his comfort zone that he was trying to show the way. Most of the good captains around are not only leaders of the team but they lead the way in their attitude.
"My message to Alastair is to go out and play his own way. Yes, we were talking as a group about being mentally aggressive - not necessarily trying to whack sixes and fours - but if you are mentally aggressive, your feet start moving and you put away those boundary balls when they come along. Sometimes that message can take a little bit of time to sink in I suppose.
"People like Alastair and Ian Bell have scored plenty of Test hundreds. My advice to them would be to go out and bat like you always have done. Those two guys are class players and can change their game to suit any situation. The rest of the guys can bat around them."
Bayliss did hint, though, that Joe Root would one day succeed Cook as captain. Asked whether he had ever had any doubts whether Cook was the right man to captain the Test team in this Ashes series, Bayliss replied: "The one thing you would say is: is there anyone to take his place?
"Joe Root is the vice captain and I'm sure will do a good job in the future some time. It may be a little early for him as yet but he's been given the job as vice captain to learn. I think that the decision to stick with Alastair has been justified because his captaincy in this game was fantastic."
While naturally delighted to see the side win in his first game as coach, Bayliss was keen to credit the players and the progress that had been made before he joined the coaching team. And while Paul Farbrace, the assistant coach and interim head coach for the series against New Zealand, was not named, it is clear Bayliss feels he has played a key role in the revival.
"The players have been heading in the right direction over the last series with New Zealand," Bayliss said. "They are interested in playing good, attacking cricket. I was lucky to come in at this time."
George Dobell is a senior correspondent at ESPNcricinfo