The versatile full-back, who can also play in central defence and has recently featured as a winger in Warren Joyce’s Reserves, is a Northern Ireland under-21 international.
Dudgeon was one of the stars of the Reserves’ 2009/10 Double-winning campaign, but has missed much of the current campaign due to a knee injury.
He becomes the fourth United player to head to Brunton Park Stadium this season. James Chester became an integral part of the Cumbrians' defence before he was snapped up on a permanent deal by Hull City, while Oliver Norwood made seven appearances before a thigh injury hastened his return to United.
Corry Evans duly replaced Norwood in October but, after one match as an unused substitute, was also struck by injury, and he returned to United before recently embarking on a fresh loan at Hull.
United now have 11 players out on loan until the end of the season.