The Gunners - who now trail leaders Manchester United by seven points with a match in hand - had plenty of possession, but were unable to really test Paul Robinson.
Rovers - who had lost five of the last eight matches - grew in confidence and survived the dismissal of midfielder Steven Nzonzi to earn a valuable point in their battle against relegation.
With United having coming from behind to win at West Ham earlier on Saturday, Arsenal could not afford another off day following two successive draws before the international break - but that is exactly what transpired.
It had been a positive start by the Gunners, with fit-again Theo Walcott looking lively down the right.
After four minutes, the England winger got away to the edge of the Rovers box before squaring to Alex Song and his shot was deflected behind.
Walcott was proving too hot to handle and his low centre across goal was hacked clear by Ryan Nelsen before he was then denied at the near post by Robinson.
Rovers - just a point above the drop zone - were in danger of being overrun.
However, Steve Kean's men were given some respite when full-backs Martin Olsson and Bacary Sagna collided when going for a 50/50 on the touchline, leaving the Rovers man worse off.
Samir Nasri then needed treatment for a clash of heads with Nelsen, returning with a thick bandage.
That break in play interrupted Arsenal's flow and on 23 minutes, Rovers had an opportunity when Laurent Koscielny tripped David Hoilett 22 yards out in a central position.
Brett Emerton, however, smashed the free-kick high into the stands.
Arsenal were soon back on the offensive and broke quickly when Robin van Persie fed Nasri on the overlap, but as the Frenchman cut the ball back on to the penalty spot, Jack Wilshere somehow stabbed his first-time shot wide.
Manuel Almunia had been a spectator, but the keeper was almost embarrassed when Olsson's 25-yard effort took a slight deflection and he was left out of position as the ball bounced off his gloves and rolled just past the post.
Hoilett then cut inside Sagna before darting into the penalty area only to skew his shot wide.
Walcott burst clear from a quick throw-out, but had no support and eventually dragged his shot wide from 20 yards.
Rovers sent a long ball into the Arsenal box from a free-kick in their own half, which Almunia failed to punch and Nzonzi saw his backward header drop just wide.
Again the hosts started brightly at the restart as Van Persie got away down the left to force a corner, which the Dutchman went across to take and then whipped straight across the face of goal.
The game, though, soon slipped back into the same pattern as Rovers continued to break up Arsenal's attacks with relative ease.
Frustrations boiled over again when Wilshere elected to pass back to Andrey Arshavin rather than shoot, the ball going behind for a corner, which Koscielny guided goalwards but Robinson held.
Wenger had seen enough and with 32 minutes to go, sent on Captain Cesc Fabregas, replacing Arshavin, on his return from a hamstring injury.
Arsenal, however, continued to lack a creative spark in the final third as time and again they failed to find the decisive pass.
Van Persie made space for a low shot at the edge of the area, but Robinson was down quickly.
Wilshere needed treatment after falling on his shoulder when sliding for a loose ball on the far side.
On 76 minutes, Rovers were reduced to 10 men when Nzonzi was shown a straight red card after jumping in two-footed on Koscielny.
Wilshere's quick feet saw the ball pulled back for substitute Marouane Chamakh on the penalty spot, but his shot was blocked by Olsson's brave dive.
Robinson produced a fine save from Chamakh's deflected header, but the whistle had already gone for a push by the Moroccan.
Fabregas smashed a 25-yard free-kick into the wall as Arsenal's frustration continued.
Michel Salgado blocked Bendtner's goalbound header off the line, and Van Persie nodded over in stoppage time to leave Arsenal counting a costly third successive draw.