The Egyptian motored into the Fulham box and finished past Sergio Rico to put the Reds 1-0 up, just 14 seconds after the visitors' offside goal.
The addition of Alisson Becker was a key part of this, but so was the acquisition of van Dijk from Southampton last January.
As the worldwide break now arrives, Klopp said he is happy for Naby Keita and Jordan Henderson, who both appeared as substitutes after hamstring injuries, to join up with their respective national teams.
Ryan Sessegnon was proving suspect defensively for Fulham, but he almost gave them the lead after racing onto Mitrovic's flick on, only to put the ball wide of Alisson's goal from close range after 24 minutes.
Xherdan Shaqiri added the second for Liverpool after the interval to move the Reds top of the Premier League table, albeit perhaps only temporarily with Chelsea and Manchester City also in action on Sunday.
Meg Ryan confirms she’s engaged to longtime boyfriend John Mellencamp
I throw fits, I gripe, I complain. "She just doesn't want anything to do with me, and I don't blame her". Mellencamp has three ex-wives: Elaine Irwin , Priscilla Esterlaine, and Victoria Grannuci.
What does it mean?
While the scoreline suggested a comfortable victory, the game appeared to turn on a controversial decision five minutes before half-time.
This was not a hugely improved performance from Liverpool after their poor display against Red Star Belgrade, but it was enough to see off a Fulham side bereft of confidence. They remain bottom, two points behind Huddersfield Town.
Fulham did not give up and were better defensively than in recent matches.
- Fulham have conceded 31 goals from their first 12 Premier League games this season; only Barnsley in 1997-98 conceded more goals at this stage of a Premier League campaign (35).
- Mohamed Salah has scored against 19 of the 21 Premier League opponents he's faced with Liverpool, failing only against Man Utd and Swansea.