The Whites scored early via Rodrigo, but then conceded in stoppage time in what looked to condemn them to further dropped points.
However, youngster Joe Gelhardt proved to be the hero, as he came off the bench to score the winner just three minutes later.
Meanwhile, down in the second tier, it was a weekend of mixed fortunes for Yorkshire's sides. Huddersfield Town picked up a point away to West Brom, Sheffield United were hammered by Coventry City, while Hull City and Barnsley secured draws in their respective clashes as the relegation battle continues to develop.
Speaking after the Norwich game, Whites boss Jesse Marsch said: “I feel like I played 96 minutes.
“There's a lot of emotion in the late stages of the match. I heard a lot about the magic of Elland Road, it was certainly an incredible experience. They asked me before will I remember this in two weeks. I said on my deathbed.
“The last moments of the game you really feel the energy. Throughout the game, the difference between Villa and this was we performed with confidence and aggression and quality and it brought the crowd into the game and gave them something to root for and be attached to."
Take a look at the latest news and transfer stories from Yorkshire and further afield, as the reaction to last weekend's action continues.