The Gunners went into the match needing to at least match PSV’s result against Bodo/Glimt to win Group A, and that always looked likely once Tierney lashed them in front.
There were brief scares after the break, especially when Adrian Guerrero had the ball in the net for the visitors, but Arsenal held on to secure a bye straight through to the last-16, ensuring they will not have to participate in the play-off round.
Mikel Arteta named a strong side from the off, as Gabriel, Ben White and Aaron Ramsdale all started, while Gabriel Jesus captained the side as he looked to end his seven-match run without a goal.
It was Zurich who started the match the better side, with the pitch still slightly covered in smoke following the pre-match flare display from the away end, and they had success down the Arsenal flanks.
But the Gunners gradually took control of proceedings and had a goal to show for it after a lovely move. White picked out Vieira, his shot was blocked and it fell for Tierney, who rifled a brilliant strike into the bottom corner.
The Scottish left-back was a constant threat in the opening 45 minutes and could have had a couple of assists as Vieira saw a goalbound efforts kept out on more than one occasion, while Eddie Nketiah’s powerful header was well-saved.
Jesus should really have scored less than two minutes after the restart as the ball dropped at his feet a few yards out, but he was somehow denied as the shot was turned over the bar.
Arsenal were frustrated in their attempts to find a second goal and as a result Arteta turned to his bench on the hour mark, bringing on Bukayo Saka and Thomas Partey as he looked to ensure there were no slip-ups.
Martin Odegaard and Takehiro Tomiyasu were also introduced but there were still nervy moments, with Guerrero denied by the offside flag and Bohdan Vyunnyk firing just wide from distance.
The Gunners held on though, the result all-important to ensure a slightly less congested calendar early in 2023.