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.
Forster also tipped substitute Marko Grujic's late header over, leaving Liverpool third in the table on 70 points, a point clear of fourth-placed Manchester City, who have a game in hand.
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.
(Reporting by Philip O'Connor; Editing by Toby Davis)
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