The Northern Ireland international, 21, was at Turf Moor on Monday to have a medical and discuss personal terms.
He has become the Clarets' record signing, with the Lancashire side paying £1m up front for the forward. There is a further £300,000 depending on appearances and a 20 per cent sell on clause.
The conclusion of the saga will be a relief for United, who have been angered by comments from Paterson and his agent.
He accused the club of going back on their word after initially rejecting a transfer request, following two failed approaches from Burnley.
And although Paterson has expressed regret, apologising to fans, Adkins was glad that a deal has gone through.
"I am bitterly, bitterly disappointed, first and foremost with the way Martin has conducted himself," he told the Scunthorpe Telegraph.
"He is a young boy with a lot of ability, but with still a heck of a lot to learn."