In a frantic finish to the half, Sainz clipped the outside of the woodwork and Long saved from Boro's Riley McGree and Hackney, but was powerless as Doak collected the loose ball from the second chance and squared for Azaz to make it three in added time.
Sainz should have reduced the deficit after 65 minutes when his volley shaved the top of the net before Cordoba brought down McGree as the Boro midfielder came from his blindside to try and win the ball in the box.
Conway’s spot-kick lacked the pace to beat Long, and he had only just taken his place on the bench after being substituted, before Sainz cut in from the left and beat Dieng with a glorious curling strike.
Norwich - who last lost at home to Blackburn Rovers on 5 November 2023 - sensed Middlesbrough were vulnerable but the manner of the equaliser was unexpected as full-back Kellen Fisher's near-post cross was only just kept out by Dieng but the ball rebounded into the net off his feet.
Norwich head coach Johannes Hoff Thorup told BBC Radio Norfolk:
"When a game is that hectic, with so many situations in it, then it's hard work for me from minute one to minute 90.
"We had to be very specific at half-time about what was important for us [to do], very specific with in-game communication to the players because there were so many details to put focus on.
"The moments we have when we are calm on the ball, especially in their half, the spaces we highlighted before the game opened up and we created chances, or at least got into some dangerous positions.
"It's fantastic for George [Long] that he's a vital part of the result [by saving the penalty]. I felt the second half was better for us and we had a better control and that save lifted the stadium - it was not just 11 against 11 at the end."
Middlesbrough boss Michael Carrick told BBC Tees:
"I thought we were fantastic for the most part, we showed what a good team we are and played some really good football.
"We conceded three goals, the third wasn't nice at all but the other two, he put them in the top corner from distance and if they’re going to score like that from there, there's only so much you can do [to stop it].
"I thought we were pretty much the better team throughout, [although] the momentum swung a little bit towards the end.
"That’s two games against really good, top opposition in this league and we’ve been the better team in both games.
"Today was a test in terms of going a goal down but the confidence we showed and togetherness to get through that was top-class."