The Gunners secured a hard-fought 1-0 victory on Thursday night, Kieran Tierney the surprise match-winner with a fine first-half strike that settled the contest.
Needing to match PSV Eindhoven's result away to Bodo/Glimt Arteta's side - showing seven changes from the 5-0 Premier League win over Nottingham Forest on Sunday - failed to fire.
But, after Manchester United had only been able to seal second place in their group earlier in the evening, the win means Arsenal will avoid the likes of Barcelona, Juventus and Sevilla as they receive a bye into the round of 16.
"We expected a really tough match," said Arteta.
"We watched how they changed since the new manager arrived, it was difficult to get any real momentum, we had to adapt.
"We had some big chances that we didn't put away and we made the game more difficult in the last 20 minutes.
"We didn't control certain situations and we had to dig in, which we did. We found a way to win and we are top of the group."
Arteta also insists he believes Gabriel Jesus' goal drought will soon come to an end as the Brazilian has now gone eight games without finding the back of the net.
"That's going to change," added the Spaniard.
"He is getting the chances, contributing to the team an awful lot, helping us to win matches with the way he competes for every single ball is incredible.
"It will come. He needs to be patient. He's been through these spells in the past and he needs to learn those lessons and not stop doing the things he does so well because that will maintain the level of him and the team."
Zurich had a late goal ruled out and winger Jonathan Okita felt the visitors applied themselves well.
"In the last 10 minutes we tried to push a little bit, but of course it's not easy," he said.
"Arsenal are a good team and we tried to do our best. We gave 100 per cent today and we're happy with our performance.
"We had a difficult start to the season and the fans have always been with us, supporting us, pushing us, singing all the time. Today you could see how they sang for the whole game."