After a poor first half, the Reds took the lead right after the break through Sadio Mane, and Wilfried Zaha equalized for the host with his first goal of the season in the 82nd minute.
The European champions restored their lead, after Roberto Firmino capitalized on a scramble ball from a corner-kick.
Alexander-Arnold has insisted that making a comeback at Selhurst Park shows his team’s fighting spirit and the Reds are satisfied with the important win.
“It was a tough game and this is always one of the toughest places to come, Selhurst Park, with the crowd right on top of you it’s going to be a really tough game,” the 21-year-old told Liverpoolfc.com
“They’ve got some really good players and it’s always, for some reason, really difficult coming here so we’ve done well to get the win.”
“We knew it meant a lot so hopefully we’ll be able to keep going and use this as momentum.”
“The first game back after an international [break] is always difficult because you go away, you play a different style and then you have to try to come back in.”
“You’ve got players travelling all over the world and obviously a bit of jetlag in there but, again, we’ve come back to have a few days of training and we’ve performed not at our best but we’ve got the win.”
“That’s what good teams do, find a way to win, and that’s what we have done.”
Liverpool are still unbeaten in the Premier League, having already played the likes of Manchester City, Chelsea, Arsenal, Leicester City and Manchester United.