Sadio Mane’s superb opener midway through the first half put the visitors in charge, but former Reds forward Ryan Babel levelled against the run of play with 76 minutes on the clock.
However, all three points were secured five minutes later when substitute Milner stepped up to convert in front of an elated away end after Mane had been fouled in the area.
Here are five talking points to emerge from a dramatic afternoon at Craven Cottage…
Pole position
Liverpool will spend the international break at the summit of the Premier League.