What won Liverpool the game was their performance right after they received the red card and were one man down. I think that was the moment when a lot of people thought that Liverpool is going to be destroyed within the next 20 - 30 minutes. But they survived it and preserved themselves the chance to win the game in the end. If they concede a goal when all the turmoil happened, I think the game is really over. But this way they forced Newcastle to still be cautious and of course it also needs a bit of luck to go home as the winner.
When Liverpool's coach (Klopp) made two straight substitutions at the 77-minute of play, the game's tempo began to shift in Liverpool's favor. Joel Matip and Marc Allister in place of Nunez and Jarell, respectively. When Klopp made that move, the game's energy changed since it appeared as though Newcastle was in complete control and Liverpool wouldn't even be able to manage a draw, let alone defeat Newcastle United at home.
Liverpool's first goal came at 81 minutes of play after that substitution was made, while their second came late in extra time (90+3). From the beginning of the first half (27 minutes) through the end of the second half, Liverpool played with 10 men on the field because of Van Dijk's red card