Kieran Tierney's sumptuous half-volley on the 17-minute mark was all that separated the two sides in North London, with Mikel Arteta's side riding out a nail-biting end to the game to hold off the threat of PSV Eindhoven.
Arteta made a total of seven changes for the Group A finale, and the absence of regular first-teamers was felt as Arsenal struggled to get a grip on proceedings early doors.
Zurich were not afraid to press high up the field, but Aaron Ramsdale remained relatively untroubled, and the hosts would break the deadlock in the 17th minute.
Reiss Nelson found Ben White on the overlap for the right-back to put in a cross, and Fabio Vieira's effort was blocked before the ball fell kindly for Tierney on the edge of the area.
The left-back unleashed a terrific half-volley into the far corner to give the Gunners the lead, and Arteta's men had their tails up after opening the scoring as they pressed for a second.
Eddie Nketiah had an effort beaten away by Yanick Brecher before being pulled back in the penalty area, but the striker elected to stay on his feet rather than go down and potentially win a penalty.
The striker's honesty did not pay off as the move broke down, but he was agonizingly close to doubling Arsenal's lead in the 37th minute, collecting with a great header from Vieira's corner, but Brecher got across well to save.
Jonathan Okita's deflected effort was the only save Ramsdale had to make in the first half, and Brecher was called upon again two minutes into the second, denying Gabriel Jesus from barely a yard out as the Brazilian's goalless run went on.
Arsenal's need to win became all the greater when PSV Eindhoven went 2-0 up against Bodo/Glimt, and they were handed a scare in the 69th minute when Adrian Guerrero fired home, but the offside flag immediately went up.
Arteta did not hesitate to bring on a wealth of first-team regulars, but attacking proficiency was largely absent in the Arsenal ranks, and Zurich substitute Bogdan Viunnyk drove through in the 80th minute before letting fly with an effort that flew just wide of the top corner.
Frustrations grew around the Emirates as a spate of perceived fouls from Zurich players went unpunished by referee Erik Lambrechts, who was loudly cheered by the Arsenal crowd after dropping his yellow card before booking Gabriel Magalhaes.
That foul from the Brazilian handed Zurich a chance to snatch a last-gasp equalizer from a free kick, and Nikola Katic headed just over the top before the final whistle blew.
PSV got the job done against Bodo/Glimt but will be forced to take on a third-placed Champions League side in the playoffs after finishing second, while Zurich are out of Europe entirely after finishing fourth in Group A, with Bodo/Glimt earning a place in the Europa Conference League.