The midfielder, who this summer will join Juventus, put Unai Emery's men ahead at the Emirates Stadium in the 15th minute before Lucas Torreira's effort just 10 minutes later deflected off Kalidou Koulibaly and into the net.
The hosts had several further chances to put the tie out of reach for Napoli but squandered many of those while goalkeeper Alex Meret showed his fine form to keep the rest out.
After seeing how poor of a performance his Arsenal side put in against Everton on Sunday, Emery made five changes to the team that hosted Napoli - including starting Ramsey in midfield.
And the move paid off after just 15 minutes when Mesut Ozil - one of the Gunners' most disappointing performers at Goodison Park - combined with Ramsey to tee up Alexandre Lacazette.
The striker sent the ball to Ainsley Maitland-Niles who found Ramsey with his pass before the Juventus-bound midfielder finished into the bottom corner of the net.
The opener was a lovely team goal and was a rich reward for how dominant the hosts had been up until that point. Arsenal were controlling possession and creating attack after attack, and so, it was hardly surprising when the second goal came just 10 minutes later.
This time Torreira was handed the plaudits after winning the ball back in midfield, driving forward, turning and then unleashing an effort that deflected off Koulibaly and into the goal.
Napoli were being outplayed by Arsenal, which was perfectly exemplified when Pierre-Emerick Aubameyang raced into the area and, despite being in a good position to take a shot, passed to Lacazette who completely missed the ball.
The ease with which both Arsenal forwards got themselves into goal-scoring positions showed just how overrun Napoli's midfield and defence were in the first half.
In terms of attack, the only attempts at goal that Napoli had came from defenders Mario Rui and Koulibaly, who both saw their efforts fly over, while Lorenzo Insigne blazed over from close range.
Carlo Ancelotti sent his Napoli side out early from the break and, within minutes, the visitors thought they had scored through Insigne, but the goal was rightly ruled out for offside.
Despite Napoli's renewed energy in the second half, Arsenal threatened twice in the space of a few minutes when Ramsey forced a save from Meret before Nacho Monreal saw a header from a corner kick fly high.
Meret then proved key again when he pulled out two superb stops to deny both Sead Kolasinac and Maitland-Niles following a swift Arsenal counter-attack.
With Arsenal squandering so many chances to put the game out of reach for Napoli, it felt as though the hosts could be punished should the visitors score an away goal.
And that they nearly did when Insigne's ball in from the right found Piotr Zielinski, who slid into the box but somehow sent the ball over the bar.
Ramsey could have bagged his second of the evening late on, but fired high on a night where Arsenal should have scored several but will still settle for a 2-0 win over a Napoli side, who sit in second place in the Serie A standings.
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