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?
Source: PA
Source: PA