Jurgen Klopp's side welcomed the Cottagers to their Merseyside back on the back of 10 successive wins from 10 home games in the 2023-24 season, including a 4-0 Europa League thrashing of LASK Linz on Thursday.
After unsurprisingly putting out a rejigged XI for that continental affair, Klopp reverted to a full-strength side for the visit of Fulham, who travelled north on the back of a thrilling 3-2 beating of Wolverhampton Wanderers on Monday.
It came as no surprise to see Liverpool start the brighter at Anfield, and they were rewarded when a Trent Alexander-Arnold free kick struck the back of Bernd Leno and bounced over the line.
However, Klopp's men were sloppy defensively and allowed Fulham a route back into the game through former winger Harry Wilson, who poked home through Caoimhin Kelleher's legs after being teed up by Antonee Robinson.
A 30-yard Alexis Mac Allister thunderbolt restored the hosts' one-goal advantage, only for Kenny Tete to capitalise on more lackadaisical Reds defending just before the break to draw Fulham level again.
After Darwin Nunez squandered a couple of chances to bag Liverpool's third, Klopp's men were staring down the barrel of defeat when Bobby Decordova-Reid outmuscled Kostas Tsimikas and headed home at the back post in the 80th minute.
The Cottagers' lead lasted all of five minutes, though, as Wataru Endo curled home a stunning leveller in the 85th minute, two moments before an Alexander-Arnold drive completed a remarkable Liverpool fightback.
With Fulham's third goal of the afternoon coming in the 80th minute, Liverpool set a new club record on Sunday, as the Reds have never come from behind to prevail so late in a Premier League game before.
The seven-goal spectacular also marked the first time since September 2020 that Liverpool had both scored and conceded at least twice in the first half of a Premier League game at Anfield, which was coincidentally a 4-3 win over Leeds United.
Meanwhile, Salah was denied his 150th Premier League goal on Sunday, but having nodded the ball down for Endo's strike, the Egyptian has now scored or assisted in 16 successive Premier League home matches.
Liverpool's extraordinary success temporarily moved them up to seventh in the Premier League table above Manchester City, who can knock the Reds back down to third if they overcome Tottenham Hotspur at the Etihad.
Next on the agenda for the Reds is a trip to relegation-threatened Sheffield United on Wednesday evening.