Fulham had the ball in the back of the net just 14 seconds earlier as Aleksandar Mitrovic headed in Tom Cairney's cross at a short-corner routine in front of the Kop.
However, the vistors' celebrations were cut short by the raised flag of the referee's assistant for offside - and the Reds broke at speed to counter effectively.
The Egyptian sped into the box past several Fulham defenders to slot past Rico and break the deadlock.
Salah has now scored against 19 of the 21 different sides he has faced as a Liverpool player in the Premier League.
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