Fabregas felt the win was important psychologically because Arsenal proved they can beat one of their main rivals without the injured Van Persie.
"What I've learned over the years is that a football team is not only about one player or two players, it's about the team," he said. "There are 25 players in the squad and whoever comes in can do very well as we showed at Liverpool.
Theo Walcott, who out jumped Jamie Carragher to set up Arshavin for his winner, agreed that they can still be contenders without the Dutch striker.
"We believe we can mount a challenge without Robin," said the England winger. "He is such a big player for us. He has scored so many important goals for us and his stature up front scares defenders.
"He will be an unbelievable miss, but we have got people like Andrey coming in to fill the gaps and he is fantastic as well."