Ali completed a brilliant campaign by taking two wickets in two balls as England won the fourth Test at Old Trafford by 177 runs shortly after tea on Monday's fourth day.
That gave the off-spinner an innings return of five for 69 on a day when he also finished off a valuable knock of 75 not out.
It meant Ali, who ended England's win in the third Test at The Oval with a hat-trick, had taken 25 wickets in the series and scored 252 runs as well.
In the process he became the first player in history to take 25 wickets and score at least 250 runs in a four-Test series.
England great Ian Botham, whose aggressive on-field personality was in marked contrast to that of the softly-spoken Ali, performed a similar feat twice during triumphant home Ashes campaigns in 1981 and 1985 -- but both of those contests were six-Test affairs.
For much of this series, Bayliss -- and Ali too -- referred to the 30-year-old Worcestershire all-rounder as a "second spinner".
It was a ruse that did not fool anyone, although England would not have cared if it helped eased the pressure on Ali.
"Maybe the psychology is working," Bayliss told reporters after England had completed their first home Test series win over South Africa since 1998 .
"There's no doubt he's our No 1 spinner and maybe he will become world-class.
"He has work to do but, knowing Mo, he won't leave anything to chance because he's one of the hardest workers in the nets," the Australian added.
As for Ali equalling a feat achieved by Botham -- and in two fewer Tests -- Bayliss said: "Most people are astounded (with that statistic).
"He does it in a quiet manner, he just gets the job done and is a great team man.
Monday's result also saw meant Joe Root's first Test series as England captain had ended in victory and the novice skipper was delighted by Ali's displays.
"It was great to see Moeen in particular in this series step up and put in some unbelievable performances to win games for us," said Root.
"The kidology seems to have worked," the star batsman added.
"He (Ali) is definitely improving all the time. I said at the start of this series, it's a great opportunity for guys in that bracket of 30-plus Test matches to stand up and become more senior," explained Root, with Ali now having played 41 Tests.
"He's definitely taken that on and put in some brilliant performances.
"Hopefully it's not just a one-off series and he can go on now and do it again and again."
Meanwhile South Africa skipper Faf du Plessis was had no doubts about Ali's quality.
"He's a real dangerous cricketer and every captain would love to have him in their team," said du Plessis.
"You often think he's not going to take wickets on the day and then you look up and he's got three or four, he's just brilliant and a real dangerous batter.
"He's a very good spinner and under-rated."
First Published: August 8, 2017, 10:01 AM IST