The two teams had cancelled each other out at the Liberty Stadium until Jack Wilshere and Aaron Ramsey combined to unlock the Swans defence for Serge Gnabry to open the scoring just before the hour mark.
Ramsey himself rounded off a free-flowing attack to make it 2-0 just four minutes later, leaving the home side with a mountain to climb.
Ben Davies reduced the deficit with eight minutes remaining, but the Gunners held on comfortably for all three points.
"We did well in the first half, but we defended too deep for the first goal," the Dane told Sky Sports News. "There were a lot of bodies but when you are that deep one or two quality players can create a chance like they did.
"We then lost our position. The players wanted to go forward immediately to get the equaliser and when you do that you leave space for Jack Wilshere, Mesut Ozil and Aaron Ramsey and you can have problems, they created a chance and scored the other.
"I am disappointed that for 10-12 minutes we gave it away, we were punished in a very small period of the game."
Swansea travel to Southampton next Sunday.