Juergen Klopp's side were outplayed by Scott Parker's adventurous team before halftime at Craven Cottage and trailed to Bobby Decordova-Reid's superb 25th-minute effort.
They were more like themselves after the break and levelled in the 79th when Salah beat Alphonse Areola from the spot after a rather harsh handball decision.
The visitors looked likely to go and grab the winner but Fulham held out for a well-deserved point to leave Liverpool level on 25 points with leaders Tottenham Hotspur.
Fulham are fourth from bottom with eight points.
Spurs held to draw
Tottenham Hotspur's Harry Kane scored his ninth Premier League goal of the season but Jeffrey Schlupp's late equaliser left the leaders frustrated in a 1-1 draw at Crystal Palace.
Kane struck in the 23rd minute with a low long-range shot that deceived Palace keeper Vicente Guaita as Tottenham began in confident fashion in the Selhurst Park drizzle.
But Palace responded impressively to dominate after the break and Schlupp grabbed a deserved leveller from close range in the 81st minute.
Ben Davies struck the crossbar and Guaita made stunning saves from Kane and Eric Dier as Tottenham sparked back into life late on but they had to be content with a draw.
Tottenham moved to 25 points from 12 games, one ahead of second-placed Liverpool who play at Fulham later.
