The 22-year-old, who progressed through Burnley's youth system, made 13 appearances for the Mariners before leaving Blundell Park by mutual consent this week. He has also enjoyed previous loan spells with Radcliffe and Alfreton Town.
Carson's arrival bolsters a United squad decimated by injuries during the opening weeks of the campaign – and he is registered and eligible to make his debut against Southend United on Saturday.
The Chester-born player has been allocated squad number 15.