Thiago maintained his form for the Reds by adding a second goal in as many outings after scoring against Porto in the Champions League on Wednesday to go with a pair of assists in his last to league games.
Former Saints defender Virgil van Dijk made it 4-0 to Liverpool seven minutes after the restart, converting Trent Alexander-Arnold’s corner-kick.
Southampton has now slipped to back-to-back defeats after seeing their four-match unbeaten run come to an end last week at Norwich City, and sits on 14 points from 13 games.
Liverpool climbed within one point of current league leaders Chelsea on 28 points, ahead of their tilt with Manchester United on Sunday.
Liverpool (4-3-3): Alisson; Robertson, van Dijk, Konate, Alexander-Arnold; Thiago (Oxlade-Chamberlain 59), Fabinho, Henderson (Milner 67); Mane, Diogo Jota (Minamino 81), Salah
Unused subs: Kelleher (GK), Morton, Matip, Williams, Tsimikas, Origi, Milner
Goals: Jota (2, 32) Thiago (37), van Dijk (52)
Southampton (3-4-3): McCarthy; Salisu, Lyanco, Bednarek (Redmond 46); Perraud (Walker-Peters 88), Romeu, Ward-Prowse, Livramento; Armstrong, Broja, Adams (Tella 46)
Unused subs: Forster (GK), Smallbone, Walcott, Valery, Long, Diallo
Bookings: Bednarek (11)