Arsenal took the lead in the 14th minute through Ramsey, who finished off a sweeping move down the right by calmly slotting the ball past Napoli goalkeeper Alex Meret.
They doubled the advantage 10 minutes later after a moment of individual brilliance from Lucas Torreira, who dashed through the Napoli midfield before spinning and switching the ball to his left foot and letting fly.
The Uruguayan's shot took a heavy deflection off Senegal defender Koulibaly and was registered as an own-goal.
The Gunners tried to increase their advantage as the first half wore on but they failed to capitalise despite several shots on goal from strikers Alexandre Lacazette and Pierre-Emerick Aubameyang, while Napoli had Meret to thank for numerous saves.
Carlo Ancelotti's men poured forward at the start of the second half in search of an away goal, with striker Lorenzo Insigne's effort ruled offside and Koulibaly denied moments later when keeper Petr Cech tipped his header over the bar.
The game quickly opened up and Ramsey, who will join Napoli's rivals and Serie A leaders Juventus in the close season, and Nacho Monreal created chances at the other end but were unable to outwit Meret again.
Napoli went closest to scoring when Insigne raced down the right and crossed to Piotr Zielinski, but the Pole's right-foot effort shot up over the bar.
A late header by defender Marcos Alonso and Kepa Arrizabalaga's goalkeeping heroics helped Chelsea beat a stubborn Slavia Prague 1-0 in their Europa League quarter-final, first leg on Thursday.
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Chelsea's Marcos Alonso celebrates scoring their first goal with Olivier Giroud. (Reuters)
Spaniard Alonso broke the deadlock for the west London side in the 86th minute when he sneaked behind two defenders to connect with Willian's cross and score what proved to be the winner in the first competitive meeting between the two sides.
Sarri made seven changes from the team that beat West Ham United 2-0 in the Premier League on Monday, with talisman Eden Hazard coming off the bench for the last half an hour.
Willian's fierce shot hit the bar in the first half before the hosts had a chance to open the scoring on the counter in the 69th minute when Ibrahim Traore drive into the Chelsea box and struck a shot that was superbly punched away by Kepa.
The Spanish goalkeeper then kept out left-back Jan Boril's close-range shot in the 80th minute.
Alonso's goal gave the Blues a vital away goal as they put behind them bitter memories of the stadium where they lost the 2013 Super Cup to Bayern Munich in a penalty shootout.
