Goals from Hector Bellerin, Mesut Ozil, Alexis Sanchez and Olivier Giroud sealed victory for Arsene Wenger’s side, who move into second.
Liverpool, meanwhile, are now eight points behind fourth-placed Manchester United following their win over Aston Villa.
“It will be very, very difficult now. It was game we needed to take something from. Obviously, a disappointing scoreline for us,” the Liverpool manager told BT Sport.
“Right from the kick-off I think we were too negative. We like to go forward with it but we put ourselves under a little bit of pressure.
“In the first 10 minutes they had a decent control of the game, as you’d expect sometimes when you come away from home, but after that first 10 minutes, one we started to get a rhythm and find our passes we were the better team and we created the better chances.
“We were looking to go into half time, 0-0 or in front but then we defend very, very poorly for all the goals. The first one, obviously when you concede the first one away from home it can be difficult. So we end up at 36 minutes conceding the first goal, it’s very, very difficult up to half-time and we concede two more.”
Prior to the fixture, the focus had intensified on Raheem Sterling’s decision to postpone contract talks until the summer but Rodgers insists it did not distract the 20-year-old.
“I thought Raheem was outstanding today, the kid done great,” said Rodgers.
“We were just poor defensively. As you see from our record we hadn’t conceded in our last six away from home. It’s just a disappointing day when you concede goals like that you’ll always be up against it.”