With the victory, the Reds have won seven consecutive matches to begin a season for the first time ever.
It is the third time in the club's history that they have won six straight league matches to begin the season (1978-79 and 1990-91).
On Saturday, the Reds took a 10th-minute lead through a Wesley Hoedt own goal, before Joel Matip and Mohamed Salah scored to secure the final margin of victory.
Liverpool are now unbeaten in their last 24 league matches at Anfield, a run that includes 16 wins and eight draws.
It is the club's best run at home in league play since 2009.
The win was slightly marred by an injury to star defender Virgil van Dijk, who was taken off in the second half with an apparent rib injury.
Liverpool are currently alone in first place in the Premier League, but can be joined by Chelsea if the Blues defeat West Ham on Sunday for their sixth straight league win to begin the season.