This was not how Arsène Wenger had intended to celebrate his 64th birthday. Borussia Dortmund had been advertised as the acid test of Arsenal's recent prowess: the first team they have faced this season who are steeped in top-level know-how.

For much of a tight but quirkily gripping encounter Wenger's players held their own against one of the continent's elite, even though they failed to find their offensive groove. And there was encouragement to be found in the way they recovered from a shaky start to stretch the visitors.

Olivier Giroud was their star turn, gilding a muscular performance with the equaliser, and Arsenal almost nicked it when the substitute Santi Cazorla rattled the woodwork in the 69th minute from Mesut Özil's pass. It was one of those evenings when the margins were slim to non-existent.

But Dortmund did not reach last season's Champions League final by accident. There is a non-stop energy about their game, along with a cold-eyed ruthlessness that is epitomised by the striker Robert Lewandowski. They staged what could be described as the classic smash-and-grab to floor Arsenal and, rather abruptly, imperil their Champions League progress.

It was a vicious counter-punch at the bitter end and Wenger will rue the manner in which his team left themselves exposed when a draw was looking like a decent result. Lewandowski did not stop to ask questions. From the moment he moved on to Kevin Grosskreutz's cross, in yards of space, there was no doubt what would happen next. He guided his volley past Wojciech Szczesny without his heart skipping a beat.

Lewandowski had been at the centre of controversy in the 64th minute when he flung out his elbow in an aerial challenge to clatter Laurent Koscielny. He was booked and, for a second, it looked as though the card from the Swedish referee, Jonas Eriksson, might have been red.

Lewandowski kept swinging until the last and, for the first time in 13 matches, Arsenal have tasted defeat. Their group, arguably the toughest in the competition, has come to feature three teams on six points, with Arsenal having played two home ties. It is likely they will need a result in either Dortmund or Naples.

Arsenal have 19 points at the top of the Premier League after eight matches but the pessimists among their support point out that they took all 24 from the corresponding fixtures last season. They are now into a testing two weeks or so, which include league games against Liverpool and Manchester United, plus the return in Dortmund. They must bounce back at Crystal Palace on Saturday.

Arsenal handed Dortmund the initiative at the outset and the goal was a calamity for Aaron Ramsey, perhaps the first black mark on his season. After Marco Reus had given the ball away to Mikel Arteta on a Dortmund thrust, Ramsey took over. But he dwelt on the ball and was robbed by Reus. The winger found Lewandowski, who shifted the ball to Henrikh Mkhitaryan, and his swerving, low shot deceived Szczesny. What Mkhitaryan lacks in vowels, he makes up for in quicksilver threat.

Jürgen Klopp, serving his touchline ban, was out of his seat in the stands while the travelling Dortmund fans, a bouncing, yellow, relentless mass, savoured the moment. One feature of the evening was the TV director's love of the quick cut between Klopp, in funky black tie, and Wenger.

Arsenal responded in style and it was Giroud who led the charge. The No9 was in swashbuckling mood, barging over opponents and rampaging in behind the Dortmund defence. He was crudely checked by Mats Hummels on the fringes of the area – the centre-half was booked – while, from the other flank, his cross was half-cleared by Marcel Schmelzer and Tomas Rosicky was able to shoot. Roman Weidenfeller was beaten but the covering Hummels cleared from the line.

There was a scare for Arsenal when Jack Wilshere needed lengthy treatment on his right foot after a heavy landing but there was delight when Giroud scored the equaliser, his seventh goal of the season. Once again there was a defensive aberration when Neven Subotic lunged at Bacary Sagna's right-wing cross, only to succeed in distracting Weidenfeller, who fumbled the ball. Giroud profited from point-blank range.

Wenger and Klopp had matched each other with their formations and it became a battle of wits in midfield. Dortmund were slick in patches but Arsenal need not beat themselves up about their game. They looked the likelier winners in the second half, particularly after Cazorla replaced Wilshere, whose evening was more of a battle against his thresholds of pain. He felt a few heavy challenges and he seemed to be attempting to run off knocks for long periods. A big bag of ice was pressed to his foot when he took his seat on the bench.

Chances were scarce, yet it was tense and riveting. One mistake or moment of quality would be decisive. Arsenal fashioned the latter when Özil, who roamed but generally struggled to impose himself, exchanged passes with Ramsey to cut-back for Cazorla. The Spaniard's first-time rising drive hit the angle of post and crossbar. Cazorla also saw another effort blocked.

If Arsenal were not to win, then they had to insulate themselves against defeat. However, Lewandowski brought them to their knees. The extent of the pain will become clearer in the coming weeks.