Sadio Mane scored the opener for Jurgen Klopp’s side in a 2-1 win, having scored twice as the Reds beat Chelsea on penalties in Istanbul on Wednesday night. Roberto Firmino added a second, before Adrian’s error gifted Saints striker Danny Ings a goal late on.
Teemu Pukki scored a hat-trick as last season’s Championship winners Norwich City bounced back from their opening-day defeat by Liverpool to record a 3-0 win over Newcastle United.
The Finnish striker, now with four goals in two Premier League matches, produced a stunning first-half volley before securing the Canaries’ first points with two more goals after the interval. Jonjo Shelvey claimed a late consolation for Steve Bruce’s side.
Play-off winners Aston Villa’s wait for a first win of the season goes on, however, as they lost 2-1 at home to Bournemouth.
Joshua King’s second-minute penalty and a debut goal for Liverpool loanee Harry Wilson put Eddie Howe’s visitors in control, though Douglas Luiz’ stunning strike from range handed Villa a second-half lifeline.
Brighton’s summer signing Leandro Trossard cancelled out Javier Hernandez’ opener as the Seagulls earned a 1-1 draw with West Ham at the Amex Stadium.
Trossard had earlier seen a first-half goal ruled out by the video assistant referee for offside, but levelled just four minutes after the Hammers’ 61st-minute opener.
Earlier on Saturday, Pierre-Emerick Aubameyang scored the winning goal as Arsenal beat Burnley 2-1 at Emirates Stadium.
The Gunners initially led through Alexandre Lacazette’s opener on his return from injury, but Ashley Barnes levelled for the visitors before half-time.
There are two games on Sunday, as promoted side Sheffield United host Crystal Palace in their first home game (14:00 BST) and Frank Lampard’s Chelsea seek to avoid a third defeat in eight days as they welcome Leicester City to Stamford Bridge (16:30 BST).