Experienced midfielder Milner was brought on by Jurgen Klopp to help see out a 1-0 victory, only for him to send a mindless clearance skywards, which Virgil van Dijk and Alisson left for one another and allowed Ryan Babel to cancel out Sadio Mane's 26th-minute opener.
That was Mane's 11th goal in his past 11 outings and the Senegal forward was brought down by Sergio Rico after the Fulham goalkeeper spilt Mohamed Salah's shot to hand Milner his shot at redemption.
Joel Matip miscued a free header from Trent Alexander-Arnold's 18th-minute corner, leaving it to the Reds' form man to open the scoring.
Mane darted towards a backpedalling Timothy Fosu-Mensah and slotted a pass to Roberto Firmino, continuing his run to convert the Brazil forward's cutback from the byline.
Fosu-Mensah made a timely tackle on Mane in the 52nd minute but the loose ball fell to Andy Robertson, whose cross was headed over by Georginio Wijnaldum – the start of a bombardment from the wings that the hosts just about survived.
Around the hour, a downpour hit Craven Cottage and Fulham belatedly entered the contest as an attacking force – Floyd Ayite having a header chalked out for offside.
Mane glanced a 72nd-minute corner against the bar before Van Dijk contributed to an improbable leveller from his fellow Netherlands international and ex-Red Babel.
But like Alisson before him, Rico erred and Milner drove a redemptive spot-kick down the middle of the goal.
What does it mean? Title race set to go to the wire
Liverpool are back on top, while City's game in hand is the Manchester derby at Old Trafford towards the end of next month. There is a lot of ground to cover before then, not least Champions League quarter-finals that Klopp and Pep Guardiola enter with their teams as favourites. Nerves and resources will be frayed to breaking point after the international break.
Mane's star quality shines through again
Five days on from tormenting Manuel Neuer and his Bayern Munich team-mates at the Allianz Arena, Mane was again Liverpool's most potent attacking threat as he created and converted the first goal before winning the decisive penalty. Handily for Klopp, he has a forward in the sort of dead-eyed form which Salah reached at this stage of last season.
Goalkeepers, look away now!
Fulham's equaliser would never have happened without Milner's unusual contribution and a defender of Van Dijk's authority might have been expected to extinguish the situation. Nevertheless, Alisson was woefully passive in allowing Babel to round him and tap into an empty net with embarrassing ease. All Fulham needed for a point at that stage was a safe pair of hands. Alas, Rico was not in possession of one.
Key Opta facts
17 - Sadio Mané has scored 17 goals in the Premier League this season, overtaking Demba Ba as the Senegalese player with the most goals in a single campaign (16 in 2011-12). Lions. pic.twitter.com/Z76rOGmelK— OptaJoe (@OptaJoe) March 17, 2019
- Liverpool extended their unbeaten Premier League run to 10 matches (W6 D4 L0) and went back to the top of the table for the first time since March 1.
- Fulham have now lost 25 of their last 26 Premier League matches against the ‘big six’ (W0 D1 L25), losing all 11 this season.
- Liverpool’s James Milner extended his Premier League record of never losing a game in which he’s scored – it now stands at 51 games (W40 D11 L0).
- Sadio Mane has scored nine Premier League goals in 2019 for Liverpool – the joint-most in the competition along with Sergio Aguero.
- Roberto Firmino’s last seven Premier League assists for Liverpool have been for either Mohamed Salah (four) or Sadio Mane (three).
- Scott Parker became the first Fulham manager to lose his first three Premier League games in charge since Roy Hodgson in January 2008.
Liverpool face arguably their most testing remaining Premier League assignment when they host Tottenham, while there is no let up for Fulham after the international break as they host Manchester City.