The 2017-18 EPL top scorer made the most of a low cross from Andrew Robertson, putting the ball home with a right-foot close-range touch.
Eight minutes after the break, he struck again, prevailing over Polish keeper Lukasz Fabianski in a one-on-one situation.
Substitute Daniel Sturridge scored the fourth goal in the 88th minute, just seconds after taking the field.
Thus, the Jürgen Klopp-coached side sits on top of the EPL table with three points, pending the Arsenal-Manchester City clash, while West Ham fell to the bottom.
The German coach's lineup included two of his new acquisitions: Brazil keeper Alisson Becker and Guinean midfielder Naby Keita, while Swiss forward Xherdan Shaqiri and Brazilian player Fabinho started on the bench.
"A very good start, for sure. Yes, it's a start - perfect as well because we won the game. For us it was not perfect, but really good. Nobody knows exactly how well you did or how worthy your pre-season is," Klopp was quoted as saying by Liverpool's website.
Klopp also confirmed that Croatian defender Dejan Lovren is facing a short spell on the sidelines due to a pelvis issue.
In other EPL action on Sunday, Southampton FC played to a 0-0 draw with visitor Burnley FC.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)