Petr Cech kept the toothless Gunners in the match too, producing three vital first-half saves as Arsenal extended their unbeaten run to 12 matches in all competitions.
Mesut Ozil thought he had stolen the points at the death, tapping home in the fourth minute of added time - only to be ruled clearly offside.
Wenger branded England's top flight a "jungle" in midweek, and here Arsenal's predatory instincts were sadly lacking.
The long-serving Arsenal boss admitted his birthday mood would be completely governed by the Gunners' result
Drawing a blank at the Emirates will no doubt have left the Frenchman hugely frustrated.
Alexis Sanchez wasted two fine chances as Arsenal missed the chance for a seventh successive Premier League victory.
Middlesbrough extended their winless Premier League run to seven matches, but still met their point with unbridled glee.
Cech's status as the man of the first half indicated Arsenal's frustrations and their fortunes to meet the interval with the game still goalless.
Mohamed Elneny blasted just wide from 25 yards, before Ozil fell in the area under Marten de Roon's challenge.
Referee Mike Dean waved away Arsenal's half-hearted penalty appeals, and after that Middlesbrough dominated.
Adama Traore picked Laurent Koscielny's pocket to race clear, only for Cech to stalk forward and produce a fine save.
The former Chelsea goalkeeper then leapt straight to his feet to deny Alvaro Negredo with the follow-up strike.
All credit was due to Cech for the double save, but former Manchester City striker Negrego should have buried the loose ball.
Gaston Ramirez whipped a free-kick against the angle of post and bar next as Arsenal continued to wobble.
Victor Valdes turned a Sanchez free-kick round a post as Arsenal punctuated Middlesbrough's purple patch.
But the Teesiders were quickly back on the attack, and again wasted a gilt-edged chance.
Cech rescued Arsenal once more, pulling off a fine save from Ramirez's point-blank header.
Negredo's effort glanced off Koscielny to fall kindly for Ramirez, only for Cech to produce another vital save.
Arsenal at least closed the half in style with their most eye-catching move to that point, Alex Iwobi racing on to a raking pass and laying off inside to Sanchez.
The Chile hitman clipped a clever pass across the box to Theo Walcott in the inside-right channel, and he duly fired square across goal for Ozil.
The onrushing Germany midfielder just arrived too late, however, leaving the sides level at the turnaround.
Sanchez missed a fine chance to open the scoring in the second half, flailing off balance with the goal at his mercy after Valdes had flapped at and missed a ball into Middlesbrough's box.
Valdes then palmed away Sanchez's wicked out-swinging 18-yard drive, before Hector Bellerin had to produce a vital sliding clearance in his own box following Nacho Monreal's rash challenge on Traore.
Middlesbrough continued to threaten on the break, with Traore drawing another fine save from Cech.
Shkodran Mustafi was perhaps a touch fortunate to escape with a yellow card for a reckless late lunge on Ramirez, upending the Middlesbrough forward after giving the ball away deep in Arsenal territory.
Sanchez spurned another fine opening when played through by Walcott after Francis Coquelin had recovered possession in the final third.
Valdes rebuffed Sanchez's effort, leaving Arsenal still agitatedly chasing that elusive breakthrough past the hour mark.
And, when Valdes headed a clearance outside his area straight to Walcott, the England forward could only scuff a tame first-time strike.
As Arsenal continued to labour for openings Middlesbrough broke at pace again, forcing Koscienly to produce a fine last-ditch recovery tackle on Negredo.
Source : PA