The Canaries appeared defeated as the visitors went into the interval with a two-goal lead, despite falling behind early in the match.
But after seeing Tommy Conway being denied what would have been a hat-trick from the penalty spot midway through the second period, the hosts found another lease of life, with Borja Sainz scoring his second to pull one back almost immediately.
The Canaries then earned themselves a point after 79 minutes when Kellen Fisher's cross was turned into his own net by Seny Dieng at the near post to restore parity.
Norwich made a bright start and grabbed an eighth-minute lead as Sainz netted his 10th goal of a highly productive campaign.
TrendingThe Spanish flyer started a move down the left flank and finished it with a brilliant strike into the top corner from well outside the box to give his side a dream start.
But then it all went wrong for the hosts as Boro turned the match on its head to take a 3-1 lead into the break.
They were level within three minutes as Conway beat George Long at his near post following an excellent left-wing cross from Rav van den Berg.
There were then chances at both ends before Michael Carrick's side scored twice late in the half to take what looked like a firm grip on proceedings.
Conway scored his second after 40 minutes following a break from a Norwich corner which saw Ben Doak race free on the left before finding Anfernee Dijksteel who unselfishly set up his unmarked colleague.
Then, in stoppage time, Long's save from a Hayden Hackney piledriver fell nicely for Doak to cross back in for Finn Azaz to apply the finishing touch from close range.
After a slow start to the second period Boro wasted a glorious chance to put the game to bed after 68 minutes when they were awarded a penalty following Jose Cordoba's clumsy foul on Riley McGree.
Conway wasted a chance to complete his hat-trick however as Long guessed the right way to save his spot-kick - and a couple of minutes later Norwich were back in it.
Once again the hero was Sainz who picked up a loose ball on edge of the area and curled it over Dieng and into the back of the net.
It was all Norwich now and they were back on level terms after 80 minutes when a mistake from Dieng gifted them a third.
Fisher's inswinging cross from the right caught out the keeper out and after getting his hands to the ball to claw it out he then inadvertently flicked it back into into the net with his outstretched leg.
Norwich were reduced to 10 men in the 86th minute when skipper Kenny McLean was shown a straight red card for a high tackle on Hackney - but they held on for a hard-earned point.
Norwich's Johannes Hoff Thorup:
To follow...
Middlesbrough's Michael Carrick:
To follow...