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.
Arsenal have now won nine on the trot and owner Stan Kroenke's public backing of Wenger has proved particularly timely given how they have responded from defeat to Aston Villa on the opening day, and the clinical edge with which they dispatched a Swansea side finding some form.
Ramsey was as key on this occasion as he has been throughout Arsenal's excellent run, and his mix of poise and composure in possession was too much for Swansea to handle.
Wenger hailed the attitude of the 22-year-old, particularly the way he has fought back from the horrific broken leg he suffered against Stoke in 2010.
"Let's not forget his career was disturbed by a big injury and you need to have patience," he said.
"After that people were impatient and put pressure on him when he had bad games, but I believe that was the turning point, when he was questioned.
"It is always an important period in the life of a footballer.
"Can you get back to basics and work harder? Or do you feel sorry for yourself and give up?
"When I saw him respond how he did I always knew he could come back, that's why I kept faith in him. He has improved his technique and does not rush his decisions, for example his goal needed a good first touch and a quick finish, and he did not rush it and showed his confidence."
The win comes 17 years after Wenger succeeded Bruce Rioch as Gunners' boss in 1996, and he admitted it made the result especially sweet.
"It's better than any cake," he said.
"The first half was difficult, Swansea played well, were sharper than us and we did not play with enough purpose.
"In the second half we played in a more compact way and took advantage every time we won the ball and had the chance to counter and be dangerous.
"We have seen a different Arsenal this season, we are more clinical and playing with more purpose. We hung on a bit in the last 15 minutes, but I felt we deserved to win.
"It's early stages, we have only played six games, and let's not forget we were in a massive crisis after losing the first game as well.
"But in our job the only way to respond is on the pitch and we did."
While Arsenal are enjoying a remarkable run of away success, Swansea remain without a home win since March, and manager Michael Laudrup felt a lack of concentration during a short spell in the second half had proved costly.
He 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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