Swansea, who are still without a league win since the opening day of the season, deservedly took the lead after eight minutes when a simple far-post corner was headed back across goal for Leroy Fer to poke home.
Klopp shies away from Liverpool title talk
Yet after the hosts dominated the opening 45 minutes, Liverpool woke up in the second half and wrestled back control of the game, levelling through Roberto Firmino's header nine minutes after the restart.
When Firmino was dragged back in the area by Angel Rangel, Milner stepped up to coolly sidefoot down the middle of the goal as Liverpool continued their excellent start to the season with a fourth successive league win.
Klopp sidesteps Sakho Liverpool storm after Hull win
James Milner scored an 84th-minute penalty as Liverpool came from behind to rescue a 2-1 win at Swansea City
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