The 20-year-old has won nine international caps and was in his country's Euro 2020 squad, coming off the bench to play against eventual winners Italy in Rome.
Levitt's sole United first-team appearance came when he started in a 2-1 Europa League defeat against Astana in November 2019.
The Old Trafford academy graduate, who joined the club aged eight, recently featured in pre-season friendlies against Derby and QPR for Ole Gunnar Solskjaer's side.
He has also gained first-team experience during loan spells with Charlton and Croatian side Istra 1961 and is looking to progress his career further at Tannadice.
Levitt said: "I am excited to be here. As soon as I heard United were interested it was a move I wanted to happen.
"I am here to help Dundee United achieve their goals but I also see this as an important part of my continued personal development."
Tannadice head coach Tam Courts added: "Dylan is a versatile midfielder who enjoys controlling games and producing assists for attacking players.
"Already a Welsh international, and part of their recent Euros squad, he's a young player with ambitions to play at the top level.
"I believe that he's capable of playing with all of our existing midfielders, but he brings something different, which will give us better options and more flexibility for the season ahead."
Levitt could make his debut in Sunday's Tayside derby against St Johnstone in Perth if international clearance comes through.