The Gunners needed to win the match and hope that Bayern Munich claimed success in the night's other Group F tie against Olympiacos, and the German champions duly obliged with a 4-0 triumph at the Allianz Arena.
Olivier Giroud was next to threaten for the hosts with a flicked header at the front post from a corner, but the Frenchman was unable to hit the target.
Zagreb began to sit back and make things difficult for the Gunners, but their stubborn resistance was broken in the 29th minute as Mesut Ozil arrived to meet Alexis Sanchez's cross, before guiding a header into the corner to give Arsenal the lead.
The goal provided a major boost in confidence for the Gunners and four minutes later they doubled their lead as Sanchez slotted Nacho Monreal's pull-back into the far corner.
Arsenal continued to look a danger in the final third and Ozil almost grabbed another goal when he danced through the Zagreb defence and forced the keeper to turn behind his powerful low drive.
Arsene Wenger's side continued where they left off at the beginning of the second half and Cazorla saw his effort tipped wide by the keeper, before Jeremy Taravel almost turned the ball into his own net from the resulting corner.
A flowing team move involving Cazorla and Mathieu Flamini set up Giroud for a chance at the near post, but the forward could not sort his feet out to turn a shot on target.
Despite Arsenal's complete dominance, the visitors showed their threat on the break when Alexandru Matel drifted inside from the left and saw his low strike deflect beyond the far post.
However, the Gunners retained their threat going forward and Sanchez headed off target, before Campbell lashed a volley wide of the post.
It seemed to be only a matter of time until Arsenal would score again, and in the 69th minute Sanchez added a third goal after racing on to Campbell's through-ball and rounding the keeper before firing a strike into the net.
Ozil then flicked a header wide in the closing stages, but the miss mattered little as the Gunners saw out the remaining minutes to claim the points and leave themselves with a chance of reaching the last 16 ahead of their meeting with Olympiacos in the final group match on December 9. body check tags ::