Liverpool could have won by a bigger margin had Rangers goalkeeper Allan McGregor prevented Darwin Nunez and Diogo Jota on either side of the break.
Darwin Nunez lost control after jostling with Joachim Andersen off the ball and thrust his head into the face of the Crystal Palace defender in the 57th minute.
Check matches and telecast details of the English Premier League 2022-23, matchday 1 fixtures.
Both Manchester United and Liverpool are yet to win a game in the ongoing English Premier League 2022-23. Get Manchester United Vs Liverpool live streaming details.
Aleksandar Mitrovic put Fulham ahead twice, threatening a major upset, but Darwin Nunez showed just why Liverpool spent big on him.
Manchester City return for the new Premier League season, which starts on Friday, as the reigning champion.
Manchester United host Liverpool at Old Trafford in the season's first 'biggest fixture' on Monday night. United have lost both the matches so far, while Liverpool have two draws.
Fans wanting to see attractive football are excited with the Premier League’s two big signings. Haaland and Nunez are classical strikers, but they also have the ability to do much more, and do it with style.
Erling Haaland's move from Borussia Dortmund to Manchester City is the probably biggest signing ahead of the English Premier League season.
It is an immediate response by Liverpool to the signing of Erling Haaland by Premier League title rivals Manchester City.
Ajax were one of three teams who won all their six group-stage matches in the UEFA Champions League 2021-22. The others were Liverpool and Bayern Munich.