Baumann came right through United's youth system, though never managed to make a senior appearance.
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He has signed a deal to 2027 with the Canaries.
Baumann told Norwich's website: "I feel great, it's a brilliant feeling. I'm really excited for this new opportunity, this is a good place to come and play football.
"I'm a central midfielder, but I can play anywhere across the midfield. I like to get on the ball in tight situations and drive forward.
"I've also represented Ukraine at youth level. A great opportunity for me, a great group of players and definitely something I'm really proud of."