The Reds were fortunate not to fall behind before Mohamed Salah's solitary chance on goal of the first half went in, as they needed a draw to advance to the round of 16.
With an open goal in front of him, Darwin Nunez, who returned to the starting lineup following injury, should have increased the lead before halftime.
However, the £64 million summer addition hit the post instead.
The Uruguayan attacker immediately made up for it, though, by precisely heading in an Andy Robertson corner just after the break.
Following a reverse feed from Salah, England Under-21 midfielder Harvey Elliott added a third goal.
