The Czech goalkeeper made 41 Championship appearances on loan at Bristol City last season and becomes Boro's seventh summer signing, joining a squad that opened the campaign with a 1-0 win over Wrexham in the Carabao Cup on Friday.
Middlesbrough have not disclosed the fee or the length of contract involved, though reports elsewhere have put the deal's total value at up to £14m including add-ons.
Vitek explained his thinking behind the move in comments to the club's official website, pointing to the ambition he has seen at the Riverside.
"The project here is very impressive, because you can clearly see the ambition and where the club wants to go," he said. "The mission is clear: to go to the Premier League.
"I think everyone at the club has that mindset, and I want the same thing."
The 22-year-old, who spent three and a half years on United's books without making a first-team appearance for Manchester United, could make his Boro debut in the Championship opener at home to newly promoted Lincoln City next Saturday, as he looks to establish himself as Middlesbrough's number one in pursuit of a return to the Premier League.