Ramsey put the Gunners in front at Emirates Stadium on Thursday and Kalidou Koulibaly deflected in Lucas Torreira's drive to double their lead.
Arsenal could have taken an even more dominant position to Naples next week, with Ramsey and Pierre-Emerick Aubameyang wasting late chances to extend their advantage.
Despite Arsenal having only won twice away from home since November, at Huddersfield Town in the Premier League and Blackpool in the FA Cup, Ramsey is confident about Unai Emery's side moving into the semis.
"We've won away this season so we're capable of doing it," Ramsey said to BT Sport.
"We've not done well of late but we're confident we can put it right, six games left in the Premier League and we want to finish the job, finish in the top four and get through to the semis.
"We've started the first leg really well so hopefully we finish the job and get ourselves in the hat for the next round."
Napoli should have claimed an away goal but Lorenzo Insigne and Piotr Zielinski missed the target, with Ramsey taking nothing for granted about Arsenal's chances of booking a semi-final spot.
"Obviously to keep a clean sheet, couple of goals, a lovely way to start the first leg," said the Wales international. "Delighted with that, we suffered a lot against a good team, we had to dig in - they keep the ball well but a great start to the quarter-finals.
"I should have scored again in the second half but these things happen, we're delighted with a clean sheet and a couple of goals going into the second leg with a little cushion. But we know it's a difficult place to go.
"It's always important to start European games with the first goal as otherwise it's difficult. They're one of the best teams in Europe and we suffered in the second half but dug in, helped each out and we're looking forward to the second leg.
"I think they were unlucky not to go through in the Champions League, they're a quality team and second in Serie A so they're tough to play against. We know from a few years ago it's difficult there but we're ready and gave ourselves a great opportunity. We need to stick together and see out the quarter-final."
8 - Arsenal have progressed from eight of their last nine European knock-out ties in which they’ve won the first leg, with the only exception coming during the 2010/11 Champions League, at the last 16 stage vs Barcelona. Halfway. #ARSNAP #UEL pic.twitter.com/T98MxcJb4I— OptaJoe (@OptaJoe) April 11, 2019