Roberto Firmino gave the Reds the lead in the opening minute as the visitors launched a speedy counter from a home corner.
After the Saints threatened an equalizer, Mohamed Salah extended Liverpool’s advantage just before halftime when he gleefully put home from a beautiful Firmino lay-off.
Salah and Sadio Mane had further chances to enhance the Reds’ lead after the break, but the two first-half goals were enough to move them to within two points of second-placed Manchester United in the table.