Steven Fletcher grabbed an equaliser before the interval after Warren Feeney had fired United in front but Goodwillie had the final say with his clinical winner.
But with skipper Chris Hogg being shown a dubious red card for a challenge on Craig Conway five minutes from time, United were able to capitalise with all three points.
Hibs lined up with young Frenchman Steven Thicot at the heart of their defence in place of suspended skipper Rob Jones while United had Darren Dods in for Lee Wilkie.
With Hibs also missing Sol Bamba and Colin Nish there was a shortage of height in the side and the visitors almost capitalised when Danny Swanson's fourth-minute corner was missed by Yves Ma-Kalambay but Andis Shala was taken by surprise and couldn't capitalise from close range.
But it was Hibs who should have taken the lead two minutes later. Derek Riordan picked out Denes Rosa's run with a perfectly weighted pass but the midfielder fired wide with only Lukasz Zaluska to beat.
United came back and Hibs had two lucky escapes when firstly Dods nodded Swanson's corner over the bar and then, from the rebound, Lewis Stevenson popped up to clear Feeney's header off the line.
However, in an entertaining match, Hibs created a fine chance as Steven Fletcher hit the byline and squared for Jonatan Johansson to flick goalwards. Zaluska did brilliantly to throw himself to his right and push the ball round the post.
Fletcher was only inches away from Riordan's cross-cum-shot but it was the visitors who went in front after 27 minutes.
Hibs had looked shaky at every set-piece they defended so it was no surprise that the goal came from a corner kick.
Swanson swung the ball in and Dods headed back across goal for Feeney to smash home a volley from six yards.
Hibs continued to look poor every time United attacked and Swanson should have done better than fire tamely over the bar.
However, against the run of play Hibs were level five minutes before the interval. Rankin played in Johansson and he delayed his cut-back brilliantly for Fletcher to lunge in at the back post and score.
United almost restored their advantage when Prince Buaben's speculative long-range shot took a wicked deflection and drifted inches wide with Ma-Kalambay stranded.
The second half was less frenetic but the visitors retained the upper-hand and were camped in Hibs' half for long periods.
Swanson wasted a chance with a 20-yard free-kick that flew over the bar before Ma-Kalambay came to Hibs' rescue with a fine save.
The ball broke to Garry Kenneth with his back to goal but he had time to spin and hit a shot on the turn that Ma-Kalambay was able to push wide with his left hand.
Hibs skipper Hogg was given a straight red card in the 85th minute for a late tackle on Conway and United took advantage with a last-gasp winner.
Zaluska's long clearance was picked up by Goodwillie and he cut inside Ian Murray before firing a low shot home and grab the three points for United.