It is understood United will receive an additional, undisclosed fee if Young wins the Serie A under Antonio Conte at Inter this season.
Young, whose Old Trafford contract only has a few months left to run, agreed personal terms with Inter last week but preferred to move immediately rather than as a free agent this summer.
United manager Ole Gunnar Solskjaer was initially reluctant to let the experienced 34-year-old leave without first signing a replacement.
Once Inter’s interest became public, Young was offered a new one-year contract to extend his stay at Old Trafford, which he subsequently rejected.
Solskjaer has now sanctioned Young’s move, with the versatile full-back searching for more regular first-team minutes.
A £17m signing from Aston Villa in 2011, Young has fallen down United’s full back pecking order behind Aaron Wan-Bissaka, Brandon Williams and Luke Shaw.
Young is reunited with former team-mate Romelu Lukaku at the Giuseppe Meazza, while United forward Alexis Sanchez is on loan with the Serie A leaders.