The Gunners needed to match the result of Group A rivals PSV Eindhoven to secure top-spot and Tierney's fine finish was just enough to see off a spirited Zurich side 1-0.
PSV won away to Bodo/Glimt so a victory at the Emirates Stadium was required as Mikel 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.
Like United at Real Socieded, Arsenal hit the front with 17 minutes on the clock as Fabio Vieira saw two shots blocked - the ball breaking kindly for Tierney to arrow home from the edge of the penalty area.
The breakthrough gave Arsenal the impetus but Vieira saw an effort blocked by an unwitting Gabriel Jesus after Eddie Nketiah seemed to be pulled back in the area, although Belgian referee Erik Lambrechts was unmoved.
Zurich forced Aaron Ramsdale into meaningful action for the first time as Jonathan Okita's shot took a deflection, the England goalkeeper reacting well to shift his weight and save low to his right.
At the other end, Tanick Brecher in the Zurich goal went one better as he tipped an Nketiah header away at full stretch.
Arsenal failed to add to their lead before the break with Jesus seeing a close-range shot turned behind by Brecher soon after the restart.
With PSV 2-1 up in Norway, Arteta called for the calvary as Bukayo Saka - an injury doubt before the game - and Thomas Partey came off the bench just after the hour.
The lead was a slender one and hearts were in mouths for a couple of seconds after Adrian Guerrero had fired home, only for the offside flag to cut short celebrations in the away end.
Arteta responded with further alterations as Takehiro Tomiyasu and captain Martin Odegaard were also introduced.
But the nerves began to creep into the home fans as Arsenal failed to add to their narrow lead, Zurich pushing forward and penning Arsenal into their own half.
Tomiyasu limped off injured to add to the late drama but the hosts eventually saw out the remainder of the contest to take their place at the top of Group A.