A Zoltan Gera cracker, Peter Odemwingie's penalty after Daniel Agger had been sent off, and Romelu Lukaku's header left former Swansea City manager Rodgers frustrated as Liverpool fans sang the name of sacked predecessor Kenny Dalglish.
Swansea showed they have got over the loss of Rodgers as Michael Laudrup's new side thumped Queens Park Rangers 5-0 away with Spanish close-season signing Michu scoring twice.
Mladen Petric also enjoyed a dream debut with a brace in Fulham's 5-0 home triumph over Norwich City while West Ham United's return to the top flight was marked by a 1-0 win over Aston Villa as fellow new boys Reading drew 1-1 with Stoke City.
Newcastle United host Andre-Villas Boas and Tottenham Hotspur later (1630 GMT) with champions Manchester City hosting Southampton on Sunday when Chelsea go to Wigan Athletic.
Manchester United and Van Persie travel to Everton on Monday.
(For more sports news and updates, follow Ahram Online Sports on Twitter: @AO Sports/ and Facebook AhramOnlineSports)