Jurgen Klopp's team bounced back from Wednesday's defeat to Chelsea with a deserved success over their Italian opponents at Santa Clara's Levi's Stadium.
Liverpool were on top in the first half, with new signing Sadio Mane seeing plenty of the ball and almost setting up Philippe Coutinho and Daniel Sturridge with crosses.
Sturridge also had a shot blocked before the break, while Adam Lallana shot over.
Firmino and Origi were among the half-time substitutes and it was Origi who broke the deadlock in the 59th minute, showing good footwork before cutting inside and shooting into the far corner.
The Belgian forward had another goal ruled out in the 71st minute for offside, before Firmino made it 2-0 two minutes later, sliding in at the far post to tap in a deflected cross from Sheyi Ojo.