The Gunners secured a 1-0 win thanks to Martin Odegaard's free-kick and a resilient display from Aaron Ramsdale, but it was somewhat soured by clashes at the end.
It is unclear what started the trouble, but both sets of fans appeared to be taunting each other and drinks were thrown across the barrier.
Stewards raced in to defuse the situation, which began after the visitors had come over to thanks the supporters who had made the journey from London to Lancashire.
According to Football London, the trouble started when the home fans began throwing bottles at the travelling Arsenal fans.
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The FA will now reportedly investigate what happened in the incident and establish further details.
Norway international Odegaard curled home the only goal in the 30th minute to further ease the pressure on Gunners boss Mikel Arteta.
Arteta's side clinched a second successive top-flight victory, but they were indebted to a controversial VAR intervention, which Burnley felt robbed them of a second-half penalty.
Referee Anthony Taylor initially pointed to the spot when Burnley substitute Matej Vydra went down under Arsenal goalkeeper Aaron Ramsdale's challenge, only to change his mind after watching the replay.