The meeting, held at the Anfield Stadium, ended 3-1 in favor of the home team.
The victory for Liverpool was brought by the goals of Joel Matip and Mohamed Salah, who scored a double.
Arsenal replied with a goal from Lucas Torreira.
In the next round, Liverpool will play against Burnley away, and Arsenal will host Tottenham Hotspur in their field.
Earlier it was reported that Manchester United lost to Crystal Palace in their third round of the Premier League.