United, who said earlier they had agreed to sign Germany midfielder Bastian Schweinsteiger from Bayern Munich, have been on the look out for a new right back after winger Antonio Valencia filled the role last season.
"Thanks Darmian, they were four great years," Serie A side Torino said in a statement as they disclosed the sale to United.
The Turin club said the player went with their blessing because it was such an "extraordinary new experience".
No figures were given but media reports have put the fee at around 13 million pounds.
The 25-year-old, a late bloomer who began his career at AC Milan, has 14 Italy caps and played in all three of their group matches at last year's World Cup.
Darmian is a solid if unspectacular full back who looks calm under pressure at both ends of the pitch.
United have also brought in Dutch forward Memphis Depay for the new season as Robin van Persie looks set to leave for Fenerbahce.
Louis van Gaal's side finished fourth in the Premier League last term and are investing heavily for a second straight close season as they look to challenge at the very top again.
(Writing by Mark Meadows, editing by Tony Jimenez)
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