And Firmino ensured Liverpool set a club record for the most points after 15 games of a league season, tucking in from close range after Van Dijk turned a Trent Alexander-Arnold free-kick back into the six-yard box.
Gomez will have scans on an injured left ankle following a full-blooded challenge from defender Ben Mee.
Liverpool fell behind but within 15 minutes they were in front, Van Dijk playing a key role in their victor.
Liverpool boss Jurgen Klopp wanted more protection for his players from the officials in the win at Burnley - a game in which Joe Gomez was injured. We have to see, we don't know in the moment.
"I'm pretty sure I don't make a lot of friends, but that's not my job", Klopp told a news conference.
"I think the referee has to make sure that things like this don't happen constantly". That was hard. You get the ball, nice, but it's like bowling because you get the player as well.
Key to their form is Ryan Fraser and Callum Wilson, with the England global Wilson scoring eight goals, while Scottish playmaker Fraser has found the net four times and provided seven assists. You try something, and you need someone to say: 'Be careful'.
And this Liverpool supporter might have regretted his tweets during the game: But fair play - a promise is a promise. Aggression is a part of football. But with six or seven yards and then (slide) these times are over.
Burnley could have been two goals to the good before Liverpool even got on the scoresheet.
Following his 300th game in management, Dyche played-down suggestions of a spat with Klopp at the final whistle, insisting: "I just said to him "well done".