Dean Ashton pulled a goal back on his home debut, substitute Leon McKenzie scored a third in the 90th minute and Adam Drury headed the equaliser in injury time to reward the home fans for their vocal support.
Norwich had shaded the first half with Damien Francis putting the home side ahead only for Jimmy Floyd Hasselbaink to equalise.
The visitors' second-half goals came from set pieces. Franck Queudrue scored twice from Stewart Downing corners and Hasselbaink beat Robert Green with a 25-yard free-kick.
The home side started brightly, but after six minutes Boro broke quickly out of defence and Green did well to block Downing's fierce drive.
Two minutes later Hasselbaink was only inches wide with a low left-foot shot.
In the 14th minute Ashton's volley flashed a yard wide of the Boro goal following good work on the Norwich left by Darren Huckerby.
Three minutes later Jim Brennan's left-wing cross reached Huckerby beyond the far post. His diagonal shot was kept out by Mark Schwarzer but was turned into the goal by Francis from close range.
Queudrue became the first of three Boro players to be booked when he fouled Ashton after 21 minutes and Ray Parlour followed him into the book for a two-footed lunge on Drury.
The Boro equaliser came in the 34th minute when Downing's shot was deflected past Green by Hasselbaink.
Boudewijn Zenden's shot brought Boro a corner in the 49th minute and Queudrue timed his run to perfection to put the visitors in front with a firm near post header from Downing's inswinging right-wing flag kick.
Anthony McMahon was booked for throwing the ball away after fouling Huckerby and Queudrue then put Boro further ahead when Downing's left-wing corner was headed on by Gareth Southgate and the full back found the net at the far post with a crisp shot from eight yards.
Norwich then brought on three substitutes - Gary Holt, Paul McVeigh and McKenzie - and Norwich seemed to step up a gear. But Downing created more panic in the City defence with two corners and a free kick.
Craig Fleming was harshly penalised for handball and Hasselbaink beat Green low to his right with a trademark free-kick.
Ashton pulled a goal back with ten minutes of normal time remaining when he beat Schwarzer to Huckerby's cross and then blasted the loose ball into the net from close range.
McKenzie made it 4-3 in the 90th minute when he raced onto Huckerby's diagonal cross from the left, his firm header giving Schwarzer no chance.
Huckerby was then fouled 20 yards from goal. Ashton's goalbound free-kick was deflected for a corner and when Huckerby's left-wing flag kick came over Drury headed in from six yards to give Norwich a hard-earned point.
After the game a shell-shocked Steve McClaren said: "This was one of those strange things that happen in football. We have lost a wonderful opportunity today.
"We were in total control at 4-1 so it was one of those things you can't explain.
"We just had difficulty in dealing with high balls into the box in the last quarter of an hour."
Canaries manager Nigel Worthington said: "At 3-1 down I thought our performance was very good. We conceded goals from two corners because people did not pick up the players we were supposed to be marking.
"The fans got behind the team and the three substitutes played well. Dean did well considering that he was playing against an England international and a Dutch international."
MAN OF THE MATCH: Darren Huckerby (Norwich City) – His never-say-die attitude epitomised the Canaries spirit. He played a part in all four goals and was a thorn in the side of the Middlesbrough defence throughout.