Hibs forward John Rankin blocked Jon Daly's first-half diving header on the line but spurned the opportunity for further heroics as his 65th-minute penalty was saved by Dusan Pernis.
And Hibs' misery was compounded when Goodwillie latched on to a loose ball and visiting goalkeeper Mark Brown was caught in no-man's land as the United striker tucked in.
The result means Colin Calderwood now has two losses in two matches in charge of Hibs, who now have two wins from 13 games in all competitions this term ahead of next weekend's Edinburgh derby with Hearts.
United, meanwhile, ended a run of three straight defeats with a fortunate victory.
Hibs' last visit to Tannadice in May resulted in a 2-0 win and confirmed a fourth-placed finish and European qualification.
How different it is now after a poor start to the campaign which resulted in Calderwood succeeding John Hughes as boss.
The former Scotland defender oversaw the 4-2 loss at Aberdeen last week as Hibs lost four goals before replying late on.
United were eager to bounce back from three losses in succession - including two in four days this week to Motherwell - and their last home outing was the controversial 2-1 defeat to Celtic, the fallout of which continued until yesterday.
Lewis Stevenson thrashed a left-footed shot wide after six minutes before Paul Hanlon timed his tackle well at the other end after going to ground inside the penalty box to challenge David Goodwillie.
Rankin shot over for Hibs before United had the best opportunity of the half after 19 minutes.
Daly's diving header from Barry Douglas' corner hit Rankin on the line before being held by Hibs goalkeeper Mark Brown.
Derek Riordan shot wide before seeing an attempt held by Pernis.
Garry Kenneth, Daly and Gomis had opportunities before Riordan spurned Hibs' best chance of the opening period.
After strong work from Galbraith and Edwin de Graaf, the striker only lofted the ball into Pernis' arms as he tried to curl a right-footed effort into the top corner.
Riordan shot straight at Pernis again, but it was the hosts who finished the half the better with three chances in quick succession.
Douglas cracked a left-footed shot from the edge of the area straight at Brown, Goodwillie's turn and shot was cleared before Daly's header fell comfortably for the Hibs goalkeeper.
Douglas and Galbraith were booked for fouls on each other in quick succession early in the second period before Goodwillie shot into the side-netting when he should have done better after being played through by Daly's knock down.
Stevenson glanced a header wide for Hibs and Douglas shot a free-kick at Brown before the visitors were awarded a penalty when Buaben fouled Galbraith.
Rankin hit the resulting spot-kick left-footed, but Pernis pushed it away diving down to his left and the Hibs forward failed to connect with the rebound.
Hibs' misery was complete when Goodwillie pounced on a long ball.
Brown and his defence mis-judged the bounce, allowing Goodwillie to knock the ball between the Hibs goalkeeper's legs and run through to make sure by prodding in.