Mane put the Reds up in the first-half, before United pegged them back just before the interval with a goal against the run of play.
Liverpool dominated much of the game and got their just reward when substitute Xherdan Shaqiri scored twice to give the Reds a 3-1 victory at Anfield.
However, Mane and Van Dijk, both who were superb at Anfield, say that the Reds fought well against their counterparts and deserved their victory.
Video from Sky Sports
"We are very happy, we played great football today from the beginning till the end. We created many chances. We deserved to win tonight," Mane said.
Van Dijk was equally emphatic with his verdict: "We had the bigger chances. They were dangerous on the break but we obviously knew they are very good at that.
"I think in the first half we were going a lot through the middle and they were winning the balls which was dangerous at times. But we dominated the game, so I'm very happy."
Watch Mane and Van Dijk's full interview above.