The 19-year-old goalkeeper arrives at the John Smith's stadium, after United opted not to offer him a new deal, putting pen to paper on a one-year contract, running until June 2017.
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As part of the deal, the Terriers have the option to extend his stay for a further year.
Dorrington is not the only 'keeper to have left Old Trafford this month, with Spain international Victor Valdes now a free agent following an 18-month spell with the 13-time Premier League champions.
Dorrington's new club begin their 2016/17 Championship campaign against Brentford on Saturday August 6.