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.
1. United eye Baggies stopper
Manchester United have been tipped to make a move for West Brom goalkeeper Sam Johnstone, as they look to find a replacement for want-away stopper Dean Henderson. The pair have both previously been involved in the England senior set-up. (Daily Star)
2. Hammers leading Worrall race
West Ham are said to be leading the likes of Everton and Brentford in the race to sign Nottingham Forest defender Joe Worrall. The club are said to want up to £15m for the towering centre-back, who has been on the books since his youth career began back in 2011. (Daily Mail)
3. Cooper reveals previous Davis interest
Nottingham Forest boss Steve Cooper has revealed he was keen to sign current loan star Keinan Davis while managing at Swansea City previously. He also praised the player’s technical skills, and claimed he has a bright future ahead. (Nottingham Post)
4. Lilywhites takeover edges closer
Preston North End are moving closer to a £40m takeover, according to reports. It has been suggested that American businessman Chris Kirchner is closing in on a deal to complete the purchase of the 1938 FA Cup winners. (The Sun)