In-form Ramsey teed-up Serge Gnabry for the 58th minute opener, before scoring his eighth goal of the season just four minutes later to seal the win and Arsenal's 12th consecutive victory on the road in all competitions.
Swansea gave as good as they got during a tight first half, and Ben Davies pulled a goal back late on, but it proved a mere consolation.
The Swans remain without a home win since March and Laudrup said: "We did well in the first half, but we defended too deep for the first goal, 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."
Laudrup also defended his decision to leave record signing Wilfried Bony on the bench, despite the striker already having bagged five goals for the club.
The Dane, who went with Michu up top, said: "It's logical, we needed players in midfield. When you face Manchester United, Liverpool and Arsenal they have good midfield players so you need players in that area."
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- Serge Gnabry