The Reds managed to win their first game on Arsenal soil since February 2000 with a 2-0 result, thanks to an Aaron Ramsey own goal and a late Luis Suarez strike, and Carragher said it was just desserts for a solid performance.
"They are such a great side and have been for a long time and it is a difficult place to come but we deserved (the win) today," he told Sky Sports.
"To be fair to them they have a lot of injuries and they had a couple more at the back today so when you have that many it is going to be awkward but I think (Thomas) Vermaelen was outstanding, especially in the first half."
And Carragher added that it was great to fight back following last week's 1-1 home draw with Sunderland.
"Last week was a bit of a comedown after everyone had started well - the players, fans and staff were on a bit of a downer and I think we had a little to prove here after that," he added.