Arsenal looked to be cruising to an unexpected victory after taking a first-half lead against the run of play through captain Robin van Persie despite being far from their best following considerable changes to their squad during the transfer window.
Perisic, last season's top scorer in Belgium, had other ideas, however, snatching a point with a spectacular volley for the 1997 Champions League winners eight years after their last appearance in the competition.
"I think that this goal was just reward for us," Dortmund coach Jurgen Klopp told reporters. "Sometimes in football you play very well but are punished if you can't score. But despite going a goal down we remained calm, we were fresh and courageous and in the end we were rewarded."
The English club had to endure a barrage of attacks and were lucky not to concede a goal before captain Van Persie struck on 42 minutes, firing in after intercepting a pass from Dortmund captain Sebastian Kehl.
With Arsenal coach Arsene Wenger watching the frantic start from the stands due to a suspension, each team missed a glorious early chance but after that Dortmund took control with their lightning-quick game.
The London side's Gervinho sneaked into the box but failed to chip the ball over keeper Roman Weidenfeller and in the ensuing quick break Kevin Grosskreutz fired over the bar.
Arsenal's back line looked far from confident despite the addition of Germany international Per Mertesacker in late August and Dortmund came close again when Shinji Kagawa foiled the offside trap but failed to hit the target from 16 metres out.
Teenager Mario Gotze, who was reportedly an Arsenal target during the transfer window, showed why he is considered the biggest talent in German football when he sliced open the defence with a clever flick to send Robert Lewandowski through but the Pole's effort was cleared on the line by Bacary Sagna.
The visitors, who lacked any real offensive momentum, struck against the run of play and briefly silenced the 65,000 crowd when Van Persie intercepted a pass from Kehl and Theo Walcott sent the Dutchman through to fire in for the lead.
Dortmund desperately searched for an equaliser in the second half, coming close with Marcel Schmelzer and Neven Subotic before Croatian Perisic timed his volley to perfection and sent the ball into the top corner to trigger wild celebrations.
"I think it was a good point because it is not easy to take some points in Dortmund," Arsenal's Sagna told Sky Television. "They are a good team. We were expecting this kind of game, we had it, and I think we can be happy with this point."
In the other Group F match, Olympique Marseille beat Olympiakos Piraeus 1-0 in Athens.
