The Senegal international scored a stoppage-time header as the Reds snatched a 2-1 win to cap a stunning finale for Jurgen Klopp’s men.
Andrew Robertson had nodded in an 87th-minute leveller after Trezeguet’s first-half opener for Villa.
The victory maintained Liverpool’s unbeaten start to the season and kept them six points clear of Manchester City at the top.
“It’s always going to be tough but our reaction was perfect,” said Mane. “In the first half it wasn’t our best performance but, like we said, we just had to push hard as a team and give everything and try, try again.
“That’s what we tried to do and at the end we created some chances and we scored two goals. For sure it wasn’t our best performance but we deserved the three points.
“I’m struggling to describe how happy we are. But for sure we are very happy and we deserved to win these three points, which was really important for us.”
Liverpool have not lost in the top flight since a 2-1 defeat at Manchester City in January.
Saturday was the closest the Reds have come this season to surrendering their unbeaten start as Villa gave them a major scare.
Manchester City also came from behind to win 2-1 against Southampton ahead of their crucial trip to Anfield next Sunday.
Mane told the club’s website: “We have big respect for Manchester City. They are a very good team but we are Liverpool and we are also a very good team. It will be a very interesting game and we are looking forward to it.”
Villa were without captain Jack Grealish, who missed the game with a calf injury he suffered at Manchester City last week.
The midfielder is expected to return at Wolves next Sunday.
Boss Dean Smith added: “He will be fine for next week.
“He trained with us on Friday but woke up this morning (Saturday) and just felt he wasn’t right.
“He had a little run on the pitch pre-game but I told him I wanted him to be 100 per cent because you have to be against teams like this.
“He wasn’t, so he made the call and I certainly supported him on that.”