Martin O'Neill's men trail leaders Hearts by five points and Glasgow rivals Rangers by two after their eighth league defeat of a miserable season.
Will Ferry and Emmanuel Agyei struck for United as Celtic lost for the second time at Tannadice this season.
The first of those reverses came during Wilfried Nancy's short-lived tenure in charge of the Hoops in December.
O'Neill has steadied the ship in two interim spells either side of the Frenchman's eight-game reign, but now faces a mountain to climb to retain the title.
Not since prior to O'Neill's arrival as Celtic manager for the first time in 2000 have they lost eight league games in a single campaign.
A boggy pitch still showing the after-effects of a rain-lashed winter did not aid the visitors as they dominated possession without any punch up front.
But the home side were worthy winners as they kept their chances of a top-six finish mathematically alive.
Ferry opened the scoring on 51 minutes after being picked out by Luca Stephenson's cross to the back post.
Agyei doubled the lead shortly afterwards from the edge of the box after more slack defending.
Celtic have won 13 of the last 14 Scottish titles but now appear the outsiders in a three-horse race.
Hearts remain on course to become the first non-Old Firm side to win the league since Aberdeen in 1985.