A day after Tim Paine, the Australia captain, said Buttler may be the best wicketkeeper batsman in the world in limited-overs cricket, Bayliss suggested "there can't be too many better". Praising his cool head and game awareness, Bayliss believes Buttler's qualities can provide an example to other players in the squad.
"How well is Jos playing? It's unbelievable," Bayliss said. "I can't really describe how good it is. Red ball, white ball, T20 - there are not too many better in the world at the minute.
"He's right up there in the top echelon. He's a bit different to some of the guys I've worked with before, but he's devastating. He can play that role he did on Sunday, in smart fashion, but when he needs to, he can pull the power out and put the foot down.
"And he's a smart guy. One of those players who looks at the situation of the game. And he's a winner. He doesn't like losing, not even in the football matches before the start. He hates to lose. Hopefully that can rub off on some of the other guys higher up the order, who themselves have played some bloody good cricket this series."
Although Sunday's century was Buttler's slowest in ODI cricket - it was the first time he had faced more than 100 balls in an innings - it was also one of his best. Showing he was far more than the instinctive player who can improvise and thrash, he marshalled an unlikely run chase from a position of near impossibility at 114 for 8. But while Bayliss accepted England's top-order had "stuffed up", he felt Buttler's contribution could provide them with an example of how to handle such situations in the future.
"We obviously stuffed up in the beginning," he said. "But I think for the first time one of the guys showed the rest how we should be doing it and how we should be approaching it in that situation.
"If we are in that situation in the future, where we lose a few quick wickets, then hopefully that approach happens a bit earlier in the order so we are only a few down, not eight.
"We still have work to do. If we didn't have work to do, we wouldn't have been 114 for 8. To me that's a bit of a reminder that there are still some things to work on. In one way it's a good reminder to the guys: don't get too comfortable with where you are, because there will be other teams out there striving to get better, and it will be difficult to stay in the form we are in.
"The good thing is we were able to win eventually from a poor position. Hopefully that sets a standard and an example to the rest of the team."