A shock was on the cards when Norwich's Michael Turner opened the scoring in the second half, but with the clock ticking down and frustration levels soaring at Emirates Stadium, Arsenal rallied.
Mikel Arteta levelled from the penalty spot, Olivier Giroud slid in from close range, and Lukas Podolski added the coup de grace to send them third in the Premier League table, one point clear of London rivals Chelsea and Tottenham Hotspur in fourth and fifth.
Everton stay sixth after maintaining their faint top-four aspirations with a 2-0 win at home to Queens Park Rangers thanks to goals from Darron Gibson and Victor Anichebe.
The picture remains bleak for QPR, who are seven points adrift of safety in 19th, level on points with Reading, who earned a point with a backs-to-the-wall 0-0 draw at home to Liverpool.
Aston Villa drew 1-1 with Fulham to move three points clear of the relegation zone, but having played two games more than 18th-placed Wigan Athletic, who were in action in the FA Cup on Saturday, beating Championship side Millwall 2-0 to reach the final.
Mid-table Southampton and West Ham United shared the spoils in a 1-1 draw.
Arsenal began the day in fifth but looked like suffering a demoralising defeat as the visitors rode their luck and then took the lead when Turner headed in an inswinging free-kick from Robert Snodgrass after 56 minutes.
Their route back into the match arrived in the 85th minute, when the linesman spotted a tug by Norwich's Kei Kamara on Olivier Giroud from a corner and Arteta slammed the penalty high into the net.
With the momentum behind them and the home crowd sensing blood, Arsenal took the lead three minutes later as Alex Oxlade-Chamberlain cut it back for Giroud to slide home with the help of a deflection off Norwich's Sebastien Bassong.
Podolski dealt the final blow in injury time, hammering home a low shot from outside the penalty area to complete a remarkable turnaround.
"I thought we wouldn't give in and fight until the end. No-one could predict we would score three goals but we have great attitude, spirit and desire - and it got us there," manager Arsene Wenger said on the BBC.
"We deserved to win but we were against a strong side and needed something special."
CLEAR WATER
After a difficult start to the season, when their top-four hopes were largely written off, Arsenal have now taken 25 points from 30. Apart from a home clash with Manchester United, they look to have a favourable run-in compared to their rivals.
They also have the opportunity to put clear water between themselves and Tottenham.
Arsenal play twice more, starting with Everton on Tuesday, before their North London rivals take to the field again, giving them the opportunity to open up a seven-point advantage as the season winds towards its climax.
"For me what is more important is to keep the run, att
