The Gunners let a seven-point half-time lead slip at Murrayfield as Stuart Hogg's second-half try set up a 20-16 win for the visitors.
Despite defeat on home soil, 25-year old Atkins believes Edinburgh can still claim the silverware when they head to Scotstoun for the second leg - which also doubles as a RaboDirect Pro12 fixture - on New Year's Day.
" It's all about next week now - that's the beauty of these games, we get another shot at them in six days' time," he said.
"We think we can win every game we play, so we'll work hard on the training ground this week and get ourselves ready for next Wednesday.
"The boys really took it to them in the first half, and I think we out-muscled them up front at the contact. But they were a lot better in the second half. We'll work on that in training and make sure we're right for the game next week."
Atkins was making his fourth start in eight appearances for the club and played a key role in a dominant first half that saw the hosts open up a 16-9 half-time lead thanks to Dougie Fife's early try.
The second half was a different story, however, as the Warriors worked their way back into the game and paved the way for the unrelenting boot of stand-off Duncan Weir to kick them back into the match with five penalties.
A Hogg sucker punch after 67 minutes then snatched the advantage before the Warriors held out for the valuable away win.
Atkins said: "As you can imagine everyone is really disappointed. We played well in patches but we made errors and let the game get away from us.
"They're a quality side and you can't give them space or they'll be on you straight away so credit to them for that.
"It was fantastic to hear that crowd out there, especially for such a big game and I'd like to thank all the fans that turned out."