Jurgen Klopp's side looked set to be heading for just its second top-flight defeat of 2019 at Villa Park, Trezeguet's 21st-minute effort having put the hosts ahead in a clash containing further VAR (video assistant referee) controversy.
Trezeguet's goal stood after a check for offside, but a review did not overturn the decision to disallow what appeared to be a valid equaliser from Roberto Firmino.
However, Liverpool's second-half dominance eventually paid off in a dramatic finish.
Robertson headed in from point-blank range before Mane flicked home the winner in the fourth minute of stoppage-time, keeping the Reds six points clear of Manchester City — which also cam from behind to win, against Southampton — at the summit.
It was the second time in a week the Reds had pulled off a late victory, added to its remarkable penalty shootout win against Arsenal in the Carabao Cup on Thursday (AEDT), where it drew level at 5-5 in the 94th minute at Anfield.