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 the 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 freekick 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, attitude and focus strong," Wenger added.
"We want consistency and that demands huge focus because you can see how quickly you're punished. We're not far away but we just have to keep winning."
QPR's unlikely hopes of staving off relegation were dealt a blow by an Everton side still entertaining an implausible late charge into the top four.
Everton took the lead in the 40th minute when a Gibson shot from distance took a wicked deflection of QPR's Clint Hill and beat Julio Cesar in goal for the Premier League strugglers.
The home team effectively ended the contest when Anichebe latched on to a knockdown from Sylvain Distin to flick home 11 minutes after the restart. Everton are now unbeaten in their last five league games.
"I think if we beat Arsenal in midweek we can talk about the Champions League but if lose we are talking about a Europa League place," Everton manager David Moyes said.
Bottom club Reading withstood a Liverpool barrage but held on for a point thanks in a large part to keeper Alex McCarthy, who was in inspired form.
Liverpool had scored in their previous 10 away league games but only once found a way past McCarthy when Philippe Coutinho had the ball in the net at the start of the second half, but it was correctly ruled out for offside.
If Liverpool lacked a cutting edge, their 35 million pounds ($53.79 million) striker Andy Carroll, on loan at West Ham United, found the net with a deflected free kick, his fifth goal in nine games, to cancel out Gaston Ramirez's opener for Southampton.
At Villa Park, the home team took the lead 10 minutes after the restart when Andreas Weimann cut the ball back for Charles N'Zogbia, who curled into the top corner. Fulham levelled when Fabian Delph put through his own net. ($1 = 0.6507 British pounds)
