With manager Arsene Wenger forced to watch from the stands due to a touchline ban, it looked as if Robin van Persie had snatched the Gunners an unlikely victory in Germany.
Having withstood a barrage of first-half attacks, the Holland international latched onto Theo Walcott's through ball to fire home three minutes before the break.
However, Dortmund continued to dominate in the second period with their brand of free-flowing football and their tactics paid dividend with just two minutes remaining.
The ball broke free to Perisic on the edge of the box and the summer signing unleashed a superb volley to equalise, much to the delight of the raucous home support who were watching their side in Champions League action for the first time in nine years.
Heading into the Group F clash, Yossi Benayoun was handed his first Arsenal start.
The on-loan Chelsea midfielder was one of three changes to the side that defeated Swansea 1-0 at the weekend, with Alex Song and Gervinho coming in for Andrey Arshavin and the injured Aaron Ramsey.
Dortmund welcomed back skipper Sebastian Kehl and 19-year-old Germany midfielder Mario Gotze after the weekend's 2-1 home defeat by Hertha Berlin, with Ilkay Gundogan and Jakub Blaszczykowski dropping to the bench.
The visitors were met by a wall of sound as they walked onto the field in Dortmund but were not put off as they attacked the famous Sudtribune.
Van Persie saw a shot blocked in the fifth minute and when the ball broke to Gervinho in the resulting play he forced Roman Weidenfeller into action.
However, Dortmund looked dangerous with their free-flowing 4-2-3-1 formation and would have netted immediately afterwards if Kevin Grosskreutz had shown a bit more composure in front of goal.
Shinji Kagawa was the next to get a shot away four minutes later, although he too blazed over the bar.
Arsenal's defence was living a charmed life as Dortmund continued to attack and only a goal-line clearance from Bacary Sagna thwarted Robert Lewandowski after he had rounded goalkeeper Wojciech Szczesny.
The Gunners looked to have weathered the storm as they began to establish a larger proportion of possession but Gotze showed his undoubted talent turning a Mikel Arteta corner into an attack that ended in Grosskreutz putting over.
Arsenal's indifferent start almost got worse when Laurent Koscielny clashed heads with Mats Hummels, although the centre-back was patched up and able to continue.
While the home side were dominating proceedings, the north Londoners were looking dangerous in their fleeting attacks and Weidenfeller was forced to save well when Van Persie broke free.
The Holland international was not be denied for long and, after winning possession on the halfway line, he ran onto a Walcott ball to fire past Weidenfeller.
Despite the visitors taking the lead, Dortmund stuck with their plan in the second period and continued to knock the ball around nicely.
Arsenal sat deep and soaked up the pressure with aplomb, although Marcel Schmelzer came agonisingly close with a drive in the 55th minute after ghosting down the left flank.
Gervinho managed to eke out an opportunity for the Gunners but they were forced to withstand an increasing number of attacks as the half wore on.
Only misplaced passing and finishing were costing Jurgen Klopp's side dear at goalscoring opportunities, much to the manager's visible displeasure.
With less than 15 minutes remaining, Arsenal brought on Emmanuel Frimpong for Walcott in a bid to shore things up as they strained to keep Dortmund at bay.
Szczesny did well to prevent Neven Subotic scoring from close range when the Serbian defender latched onto a Gotze corner in the 82nd minute.
Gotze screwed wide moments later as Dortmund pressed for an equaliser, which came in the 88th minute through Perisic's wonder goal.
The ball fell kindly to the substitute from a corner and he unleashed a volley that rifled into the top right-hand corner with two minutes remaining.
Dortmund pressed for the winner in the dying minutes, although resolute defending kept the scores level.