Jürgen Klopp’s side fought back from an early deficit at St Mary’s, with Naby Keita’s header cancelling out Shane Long’s strike for the hosts.
There was work still to do after the interval, though, and the visitors’ determination forced two late goals in front of the away supporters.
Mohamed Salah ran the length of the Southampton half and clipped in a finish to edge Liverpool in front and Jordan Henderson added gloss to the turnaround soon after.
It was another exhibition of character by the Reds and Van Dijk pointed to that never-say-die attitude as the reason for their challenge for Premier League glory.