The 29-year-old midfielder agreed a three-and-a-half-year contract and passed a medical after agreeing personal terms for the move to Parkhead.
Robson, who was United's captain, has hit 12 goals already this season, including seven in his last five appearances.
Celtic's Jim O'Brien moves in the opposite direction, joining United on loan until the end of the season.
United manager Craig Levein confirmed he and chairman Eddie Thompson had wanted to keep Robson.
Levein said: "Both myself and the chairman would have preferred Barry to remain a Dundee United player, but he made it abundantly clear on a number of occasions that he wanted to move on.
"He felt that at his age, he had one last chance of a major contract and we couldn't persuade him otherwise.
"He's a fantastic player, greatly admired by the chairman and myself and we wish him well in his future career."