Defender admits opening day draw against Sunderland left the Reds deflated and reveals Kenny Dalglish's men keen to take advantage of injury-hit Gunners
The Reds ran out 2-0 winners in London after an Aaron Ramsey own goal and a strike from substitute Luis Suarez gave them their first three points of the season.
After the game Carragher admitted that the disappointing draw at home to Sunderland in the first game of the campaign had spurred the players on to beat Arsenal.
He told Sky Sports: "Last week was a comedown after everyone had built themselves up for the start of the season and we had started great.
"Sunderland should've been down to 10 men, game over but unfortunately it didn't happen and we didn't do so well.
"The club, players and staff were on a bit of a downer so I think we had something to prove here today after that and we did that."
As for the match itself, Carragher was full of praise for Arsenal centre-back Thomas Vermaelen who was imperious for the Gunners up until they conceded in unfortunate fashion.
He continued: "They've got a lot of injuries and they had a couple of the injuries at the back today so it's going to be very tough for them and we wanted to take advantage of it.
"Vermaelen was outstanding, first half he would've had the champagne [for man of the match] if it finished 0-0 because without him I think we could have got a goal or two in the first half. But credit to us we can't worry about Arsenal we've got to worry about us."
New Spanish left-back Jose Enrique was the actual winner of the man of the match award and he was keen to insist that both he and Liverpool can improve further.
He said: "We did well because we won against one the best teams in England. I think we have a lot of new players and we need to play more together but I think we played okay [today].
"I played more in attack today because I know the players better and I think I played okay but I can improve."