Supporters from both sides clashed at the end of the game at Turf Moor, which Arsenal won 1-0 thanks to Martin Odegaard’s free-kick.
The trouble occurred in the David Fishwick Stand, which houses away and home fans.
At the full-time whistle supporters clashed in the middle of the stand, where barriers and stewards separate them.
The barriers kept fans apart and further stewards arrived on the scene to diffuse it, while the stadium announcer asked all supporters to leave the ground too.
Plastic bottles appeared to be thrown by supporters during the clashes.
The FA are now looking into the matter to understand what happened and how the trouble came to take place.
Arsenal’s victory over Burnley was their second Premier League win of the season after losing their first three games.