The reigning MLS champions squandered a one-goal lead to be beaten 2-1 at the Toyota Stadium in Frisco on Saturday night.
Gyasi Zardes put the Galaxy ahead in the 59th minute before goals from Blas Perez and Tesho Akindele gave Dallas a fighting win which Arena puts down to his own coaching shortcomings.
"We didn't do well at the end of the game, obviously the last 15 minutes, and that's my responsibility," Arena told the press after the defeat.
"When a team plays that poorly on the road, they're not well-coached.
"We blew a lot of points this week. We could have walked away, stole maybe six points this week, [and instead] we walk away with one, and that's my responsibility.
"I have not done a good job with this team, to get them tactically right at the end of games."