The Egyptian reached the landmark as the Reds avoided a season-opening loss away from home.
The Egyptian has now scored eight goals on the first day of the league, tying Shearer, Lampard, and Rooney for the joint-most.
Salah scored late in the second half to bring them back into the game after Mitrovic’s second goal put them ahead.
In the 32nd minute, a good combination between Democratic Republic of Congo attacking midfielder Neeskens Kebano and Kenny Tete on the right flank resulted in the latter producing an excellent cross towards the post where Mitrovic was waiting.
The Serbian leapfrogged Trent Alexander-Arnold and scored with a powerful header that Alisson Becker couldn’t stop.
As Liverpool piled pressure on Fulham, Darwin Nunez almost scored two minutes after the hour mark. Salah delivered a low cross that was met by the Uruguayan, who attempted to flick the ball into the net, but Marek Rodak stretched wide enough to save.
Nunez, on the other hand, scored in the 64th minute to make his debut premier league goal.
In the 71st minute, referee Andrew Madley pointed to the spot after Virgil van Dijk brought down Mitrovic, and the Serb converted successfully from the spot for his second goal.
As a result, Mitrovic is the first player since Steve Mounie for Huddersfield against Crystal Palace in August 2017 to score two goals for a newly promoted side on the first day of a Premier League season.
Liverpool fought back and tied the game in the 80th minute. Salah was set up by a diagonal pass from Alexander-Arnold. The Egyptian took advantage of the situation and equalized for the Reds, with Nunez assisting.
Nunez is only the third player in Premier League history to score and assist off the bench on his debut, and the first since Alvaro Morata for Chelsea against Burnley in August 2017 and Sergio Aguero for Manchester City against Swansea in August 2011.