Arsenal are in danger of falling into fifth place by missing out on three points, as Tottenham and Newcastle are breathing down the Gunners' necks.
Yossi Benayoun scored just after a minute of play with a well-placed shot to give Arsenal the early lead.
But the Canaries would score twice in the first half, one off a poor effort by Wojciech Szczesny, as Wesley Hoolahan and Grant Holt found the back of the net take the lead before the half-hour mark.
Robin van Persie would add two goals in the second half while Arsenal battled back. The Dutch forward scored in the 72nd and 80th minute as a magical comeback appeared to be in the works in Arsenal's final home match of the season.
However, Morison had other plans. The 28-year-old forward took advantage of some questionable defending to level the match. Though the Gunners had plenty of chances to find the game-winning score, Arsene Wenger's side simply couldn't get the job done, as their 2012-2013 Champion League hopes are in serious jeopardy with a final match against West Bromwich on the schedule.
The Gunners took 23 shots with 11 on target, while Norwich took 13 shots with four on goal. Arsenal also had possession for 63 percent of the match.
Here are the highlights from Arsenal - Norwich City: