Two goals from Dries Mertens either side of half-time gave the visitors a commanding advantage, as the Giallorossi struggled to produce the sort of performance that they managed in the away win over Inter Milan the previous weekend.
Kevin Strootman belatedly halved the deficit in the dying moments but, despite a frenzy of activity in injury time, a late equaliser would not fall for Luciano Spalletti's side.
The result means Napoli are now two points behind Roma in the Serie A table.
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The hosts were left to wonder what might have been, after a frantic finish that saw Napoli goalkeeper Pepe Reina claw a deflected Diego Perotti strike away via the aid of the woodwork.
During that final period the Giallorossi also had a penalty appeal waved away and saw a number of attempts from corners fly over Reina's goal - agonising misses in the pursuit of an equaliser.
Prior to that, however, the Giallorossi had struggled to cut open their opponents and were, in turn, cut open twice by incisive Napoli attacks.
The first goal saw Mertens steal between the defence to flick a calm shot over Szczesny, before a second-half ball from Lorenzo Insigne gave him another straightforward chance to get his second.
Roma improved after the second half additions of Mohamed Salah and Bruno Peres, with Salah hitting the post in agonising fashion in the final 10 minutes.
Strootman slotted home soon after but, with only injury time remaining, ultimately time ran out on the Giallorossi to salvage a draw.
Coach Luciano Spalletti opted to start both Stephan El Shaarawy and Diego Perotti - with Bruno Peres and Mohamed Salah dropping out of the team that won at the San Siro six days earlier.
Daniele De Rossi captained the side.
13' - Good chance for Mertens. He gets between the defenders but plants a free header over the bar. A let-off.
23' - GOA.... No, disallowed! Perotti controls Koulibaly's clearance and rifles home - but the ref blows for handball!
26' - Goal. Napoli lead. Mertens gets in behind the defence and flicks the ball beyond Szczesny. 0-1.
39' - Now Napoli have a goal struck off - Mertens turns home again but is pulled back for fouling Fazio in the build-up
See more images in our match gallery.
50' - Goal. Napoli double their advantage. Insigne finds Mertens inside the box for the close-range finish. 0-2.
52' - How will Roma respond? Perotti cuts in from the left and hits a drive that Reina is forced to save...
69' - Another chance for the hosts ... but Peres flashes a half-volley wide under pressure from the covering defender.
85' - POST! Salah gets in behind the defence and beats Reina, but his shot comes away off the woodwork...
89' - GOAL! Roma halve the deficit. Perotti feeds Strootman and the midfielder drives home! 1-2.
90' - Roma pouring forward... Salah finds Perotti but he cannot steer his shot inside the far post!
90' - BAR! Reina makes an incredible reflex stop to claw away Perotti's deflected strike thanks to an assist from the woodwork!
See more game statistics in our dedicated match centre.
“Napoli were quicker and moved the ball about better than us in the first half. We weren't able to find space between the lines or raise the tempo, which would have prevented them from taking up good positions and making it hard for us to regain possession."Napoli deserved their lead in the first half because they were better than us. It was a totally different game in the second half and luck came into it a little too."
- Luciano Spalletti
The Giallorossi now turn their attention back to European matters, as they travel to France to face Lyon in the first leg of their Europa League last-16 tie on Thursday.
Sunday then sees another away trip, as the Giallorossi travel to face Palermo in Serie A - before the return leg against Les Gones the following week.
Goalscorers: Strootman (89'); Mertens (26', 50)
Roma: Szczesny; Jesus, Fazio (Peres), Manolas, Rudiger; Nainggolan, De Rossi (Paredes), Strootman; El Shaarawy (Salah), Dzeko, Perotti.
Napoli: Reina; Ghoulam, Koulibaly, Albiol, Hysaj; Hamsik, Jorginho, Rog; Insigne, Mertens, Callejon.
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