The forward, who is joint top of the league's scoring charts this season with Chelsea's Cole Palmer missed FA Cup semi-final win over the Blues
Phillippe Coutinho returned from the leg injury that saw him limp out of Monday's 1-0 win over Watford, but the Reds struggled to spark offensively against Southampton's resolute defence.
The hosts were handed a lifeline in the second half when referee Robert Madley pointed to the spot after Southampton defender Jack Stephens handled the ball in the box, but Fraser Forster got down smartly to his right to keep out Milner's spot kick.
Later on Sunday, sixth placed Arsenal, who are 10 points behind Liverpool with three games in hand, host fifth-placed Manchester United, who are five adrift of Juergen Klopp's side having played two matches fewer.