The Reds dominated the first-half but it was Arsenal who broke the deadlock with Alexandre Lacazette able to capitalise on Andy Robertson's error to put the ball past Alisson, only for Sadio Mane to equalise three minutes later.
Lacazette had the chance to equalise for Arsenal thanks to a superb pass from Dani Ceballos, but the Frenchman struck his shot directly into Alisson.
The two teams will play each other again on Thursday in the Carabao Cup fourth-round.
Speaking after the game, Robertson said: "It was a fantastic counter-pressing game from us. We all reacted when we lost the ball. We put them under pressure and forced them into mistakes. With the ball we showed quality.
"We can't have too many complaints, other than my assist at the wrong end. 99 times out of 100 I clear that. The lads dug me out of a hole. A couple of years ago I'd have let that get to my performance.
"We showed a positive reaction. They've got a fast frontline which you need to be wary of. We dealt with them well and caught them offside a few times.
"We could have had a couple more. We'll take the 3-1. It's nice to beat your rivals and make it three wins from three."