The vice-captain calmly converted a spot-kick nine minutes from time after a foul on Sadio Mane, whose first-half goal was cancelled out when former Reds forward Ryan Babel pounced on a defensive mix-up.
It meant Jürgen Klopp’s side moved back to the top of the table, two points clear of Manchester City having played one match more than the reigning champions.
Team news
Klopp made two changes to the XI that started in Munich in midweek, with Fabinho and Adam Lallana selected to replace Jordan Henderson, who missed out due to injury, and Milner.
First half
Liverpool quickly established control of possession at Craven Cottage and Mohamed Salah’s eyes lit up when, just four minutes in, Lallana’s lofted pass over the defence was nudged into his path.
On this occasion, though, the Egyptian could only skew a shot wide of the target from the six-yard box.
Jean Michael Seri dispatched a curler from left to right too high to concern Alisson Becker at the other end, but it was a fleeting glimpse for the home side as the visitors dominated.
Andy Robertson might have done better than find the goalkeeper with a low cross after his sprint into space along the left was picked out by Joel Matip – a tactic consistently employed by Klopp’s men – and Trent Alexander-Arnold swept in a fine delivery from the other flank that evaded his teammates.
A more intricate move generated the breakthrough for Liverpool on 26 minutes.
Mane got it, having released Roberto Firmino into freedom in the left channel of the area and dashed into the middle to meet the return pass with a thudded finish beyond Sergio Rico.
The No.10 reached the 20-goal mark for the season with the emphatic conversion, equal with his tally for the entirety of the previous campaign.
Robertson saw a shot blocked from the angle after another clever give-and-go with Lallana, before Babel scampered clear of Alexander-Arnold down the left and pulled a pass back to Tom Cairney, whose strike sailed over.
And so Liverpool carried a one-goal advantage into the half-time interval.
Second half
With the weather worsening in London, the Reds continued to set the tempo in the game and Lallana’s persistence and strength in the area almost forced a second goal.
He managed to hold off a series of challenges and turn the ball to Georginio Wijnaldum; his diagonal hit across the hosts’ box was flicked towards goal by Firmino but blocked by Timothy Fosu-Mensah.
Wijnaldum headed into the stand from Robertson’s cross, with Salah perhaps better placed behind him, and Rico then thwarted Virgil van Dijk’s nod after Fabinho had hoisted a delivery to the far post.
But with the deficit only one, Fulham began to rally and had the ball in the net on 64 minutes, though Floyd Ayite was offside when he diverted a deflected shot around the reach of Alisson.
Moments later, the Liverpool goalkeeper had to get down to collect a tricky strike from the edge of the box by Andre-Frank Zambo Anguissa, following a breakaway led at speed by Babel.
With 20 minutes remaining, Klopp sent on Milner and Divock Origi for Lallana and Firmino, and from the resumption of play Mane guided a header from Alexander-Arnold’s corner off the crossbar.
It was a pivotal piece of misfortune, because Fulham soon levelled.
Milner’s clearance towards his own goal forced Van Dijk to shepherd possession back to Alisson but his pass fell short and Babel capitalised on their hesitations to tap into an unguarded net.
The Reds had to find another way – and they did.
The clock showed 80 minutes when Salah’s latest shot was fumbled by Rico, and Mane’s urgency on the follow-up drew a clumsy foul by the home stopper and earned a penalty.
Substitute Milner took responsibility from 12 yards and deftly clipped a cool finish down the middle to snatch three more crucial points in the Premier League title race.