Zirkzee agreed a five-year contract with the Premier League club that includes the option of a further 12 months.
United reportedly opted to pay slightly more than Zirkzee's £33.6 million release clause in order to spread the fee over a three-year period to help with Profit and Sustainability Rules.
The 23-year-old rose to prominence in Serie A last season as he helped Bologna unexpectedly qualify for the Champions League.
He scored 12 goals in 37 appearances for Bologna last term and was also part of the Netherlands squad that reached the Euro 2024 semi-finals before losing to England.
United's Dutch boss Erik ten Hag spoke to Zirkzee to convince his compatriot to make the move to Old Trafford.
"I am a player who has always dedicated everything to winning; I'm ready for this next challenge, to go to another level in my career and win more trophies," said Zirkzee, who started his senior career with Bayern Munich and had loan spells with Parma and Anderlecht prior to joining Bologna.
"Having had discussions with the manager and the leaders of the club I know how exciting the future is going to be here and I cannot wait to play my part in achieving success for Manchester United.
"It is a privilege to join such an iconic club. I have to take a short break now after being with the national team, but I'll be coming back ready to make an impact straight away."
Zirkzee will compete with Rasmus Hojlund, Marcus Rashford, Alejandro Garnacho and Antony for a place in United's attack in the coming season.
