Liverpool maintained their place at the top of the Premier League and their 100% start to the season as Arsenal were well beaten at Anfield.
The Gunners, the only other side in the top flight to win their first two games, had opportunities to shock Liverpool in a tight first half, especially when record £72m signing Nicolas Pepe shot straight at keeper Adrian after running clear.
Liverpool took control when Joel Matip put them ahead with a powerful header from Trent Alexander-Arnold's corner four minutes before the interval.
And any hopes manager Unai Emery had of watching his side mount a recovery were snuffed out in the 49th minute after a moment of madness from new signing David Luiz, who needlessly dragged Mohamed Salah back in the area.
Liverpool's Egyptian attacker drilled home the penalty and then embarrassed Luiz again out on the touchline at the start of a dazzling run that ended with a low, curling finish into the bottom corner just before the hour.
Arsenal substitute Lucas Torreira pulled one back late on, but nothing was getting in the way of Liverpool's 12th successive Premier League victory, equalling their best sequence in the top flight under Kenny Dalglish between April and October 1990.