The Gunners' afternoon had started so well with a goal from Yossi Benayoun after just 65 seconds. However, the Canaries, already guaranteed their top-flight status for next season, rallied and were level after Wojciech Szczesny spilled Wes Hoolahan's shot before Grant Holt's deflected effort gave the visitors a deserved half-time lead.
Arsenal rallied after the break as a brace from Robin van Persie in the space of eight minutes put the home side back in charge, only for Norwich substitute Steve Morison to grab a deserved point. There was still time for referee Anthony Taylor to miss a clear shove by Kyle Naughton in the back of Van Persie as he looked to tap in a winner at the far post before Canaries goalkeeper John Ruddy made a couple of important blocks.
Arsenal have a slender two-point lead in third place and now must hope Tottenham and Newcastle fail to make up ground tomorrow as the race for Champions League qualification is set to go to the wire.
A win looked a very safe bet indeed when Benayoun marked his 32nd birthday with the opening goal after a minute's play. He picked the ball up on the left side of the Norwich penalty area, cut inside and curled a sublime effort up over Ruddy into the far corner.
That should have settled any nerves, but instead, the Canaries were galvanised.
Bradley Johnson's chipped free-kick was headed goalwards by diminutive Canada frontman Simeon Jackson, and the equaliser came in the 12th minute.
Full-back Naughton, on loan from Tottenham, raced forward down the left and played the ball through to Hoolahan on the penalty spot. The Republic of Ireland midfielder did not really connect properly with his shot but it was enough to squirm past Szczesny and across the line.
Now it was Arsenal whose resolve was tested, as the atmosphere grew decidedly edgy.
Norwich continued to ask questions of the home defence and went ahead in the 27th minute. A quick break saw Hoolahan release Holt at the edge of the Arsenal area. As former Norwich loanee Kieran Gibbs backed off, the 31-year-old's shot caught the defender's boot and looped over Szczesny and in.
Just like against Wigan, the Gunners were shell-shocked and their afternoon took another turn for the worse in the 33rd minute when full-back Bacary Sagna pulled up with a calf injury and had to be replaced by Francis Coquelin.
Arsenal's tempers were frayed, and Benayoun was fortunate not to be punished when he kicked out at Simon Lappin after a tussle with the Norwich defender.
The visitors remained composed and felt they should have been awarded a penalty when Coquelin up-ended Hoolahan following another quick break down the right channel.
Referee Taylor and his assistants were in the thick of the action again a few minutes later when Russell Martin had his shirt pulled by Laurent Koscielny at a corner, but again the officials gave nothing - much to the frustration of the Norwich bench.
It could have been 3-1 in first-half stoppage time but for a perfectly-timed sliding challenge by Koscielny to block Holt's close-range effort.
There was more purpose about Arsenal in the second half, Van Persie rolling the ball across the face of goal following a sustained period of pressure.
Norwich, however, also looked to be positive and Jackson broke clear of Thomas Vermaelen and skipped into the Arsenal area, but his shot was straight at Szczesny when he probably should have squared for Holt to tap in.
As the hour mark passed, Van Persie failed to beat Ruddy with a tame strike from 12 yards before Aaron Ramsey dragged his shot wide. The Wales midfielder was replaced in the 63rd minute by Alex Oxlade-Chamberlain as Wenger looked to inject some much-needed impetus into his side.
It was captain Van Persie who made the breakthrough though, as he volleyed home Alex Song's excellent angled ball through the penalty area.
Arsenal maintained their momentum as Norwich fought hard to disrupt it as much as possible, with Gunners boss Arsene Wenger exchanging a few heated words with Canaries counterpart Paul Lambert across the technical area.
There was elation, though, when the ball ricocheted into the path of Van Persie, who reacted quickest to lash his shot under Ruddy.
However, the Canaries were not to be denied a famous result as with Arsenal pressing forwards, they were caught out by a ball over the top and Morison ran on to slot past Szczesny.
There was still time for Ruddy to deny Van Persie with a fine close-range block before a clear push by Naughton in the back of the Arsenal captain as he looked to tap in at the far post went unpunished - and the Gunners' Champions League destiny was again taken out of their hands.