The former England captain took to Instagram to congratulate the club's players and staff and share his excitement for the team's future.
"Very proud moment for our club today and the team did us proud," the former Manchester United and LA Galaxy midfielder said.
"It's been a long journey but this is only the beginning. To Diego (The Boss) the team and our entire staff...we should be very proud how far we have come and what the future holds.
"Exciting times ahead."
But Vela, on his 31st birthday, produced a piece of individual brilliance to give LAFC the lead on the stroke of half-time, dancing his way to the edge of the area before executing a sublime chip over Robles.
The hosts thought they had doubled the lead 30 seconds into the second half but Rossi was denied by the offside flag once again.
Mexico international Rodolfo Pizarro then came within inches of netting Miami's historic first goal in MLS, but his prodded effort fell just the wrong side of the post.