The forward opened the scoring in the second half at Turf Moor with a brilliant left-footed drive that was totally out of keeping with what had been a scrappy, tense game.
The contest continued in much the same vein after Mane's strike, and it looked like the hosts would be the ones to benefit from that fact as Johann Berg Gudmundsson equalised in the 87th minute.
However, Jürgen Klopp's side, who had enjoyed the lion's share of possession and the attempts on goal, snatched it at the death through Ragnar Klavan - deservedly, according to Mane.
"I think it is not our best game but I think we deserved to win because we created many chances and we got two goals," he told Liverpoolfc.com.
"We have some important players missing today. But we have other players and we can count on them and they do well.
"But more important we are a great team, great club and that means we are on a good way. Everybody is working hard so we deserved to win."
In beating the Clarets, Liverpool made it three wins and one draw from four fixtures between December 22 and January 1.
Mane believes the Reds are delivering on their potential but is refusing to look at the Premier League table just yet.
He added: "Honestly I think this is what we can do because we try our best and try to win [as many] games as we can. To get 10 points out of 12 I think is great.
"I think this season is too early to talk about [the table]. I think more important for us is keep [going] game by game and try to win and we are going to know the table at the end of the season."