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, but it proved a mere consolation in a competitive encounter at the Liberty Stadium.
Arsenal have now won nine on the trot and owner Stan Kroenke's public backing of Wenger 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.
But even the Wales international struggled to enlighten a first half which produced a total of zero shots on target.
There were pleasing passages of play, but little in the way of goalmouth action as the two sides cancelled each other out.
Jonjo Shelvey, back at the Liberty for the first time since his startlingly mixed performance against Liverpool, initially played a prominent part in a bright Swansea start, his range of passing helping get the hosts in motion.
The midfielder also had the first meaningful effort on goal - one of very few during the opening 45 minutes - bringing down a high ball and fizzing a fierce drive narrowly over Wojciech Szczesny's bar.
Arsenal began to find their rhythm with Ramsey delivering a pinpoint cross for Per Mertesacker to head wide, before the Welshman arrowed a low shot around a yard wide from the edge of the area.
Source: PA
Source: PA