Hibs appeared to be cruising to victory at Tannadice when Colin Nish netted twice in a blistering four-minute spell, before Craig Conway missed from the penalty spot for United.
But Garry Kenneth hauled the home side back into the match early in the second half before Goodwillie came off the bench to have the final say with a last-gasp equaliser.
United made two changes from the side who drew with Celtic in their last outing.
Warren Feeney had impressed for Northern Ireland against Poland and Slovenia and was presumably rested by boss Craig Levein when he was named on the bench along with Mihael Kovacevic.
Sean Dillon and Jon Daly both started, while Hibs made a single change following their goalless draw against Aberdeen, with Sol Bamba back from suspension to start ahead of David van Zanten.
However, Daly's game was over within a few minutes when he was stretchered off following what appeared to be an innocuous challenge by Chris Hogg and was replaced by Feeney.
Then, with his first touch of the ball, Feeney lost possession to Nish, who stunned the home side by rifling home a superb speculative shot from all of 30 yards with just six minutes on the clock.
The Hibs striker doubled his tally just four minutes later. Prince Buaben's disastrous attempt to nod back to goalkeeper Lukasz Zaluska broke to Nish to slot into the net from close range.
Only an excellent one-handed save from the United goalkeeper prevented Nish from claiming his hat-trick within a quarter of an hour when he palmed to safety after Nish was set up by Denes Rosa.
James Wesolowski picked up the first booking of the game for diving when he threw himself down in the box under pressure from Rob Jones.
At the other end, Hibs had another chance to add to their lead when Nish turned provider for Steven Fletcher, who starred for Scotland in the midweek win over Iceland.
But he just failed to connect at the far post and missed the opportunity to stab home.
United then squandered a great chance to haul themselves back into the game when referee Eddie Smith pointed to the spot after 31 minutes.
Conway stepped up for the penalty but was denied by a superb block from former Arabs goalkeeper Grzegorz Szamotulski.
There was concern for Hibs when Fletcher appeared to injury himself shortly after the restart after testing Zaluska with a tame shot.
He was quickly able to return to the action but a bigger worry was the sight of United pulling a goal back with 53 minutes gone.
Paul Dixon provided the cross and Kenneth did well to direct a header past several green jerseys and over the line.
Dixon was the provider again with a great ball to Francisco Sandaza but he somehow managed to stab wide of the upright a yard from goal.
United sensed that at least a point was within their grasp and Kenneth nodded agonisingly wide after Lee Wilkie had sent a head-flick into his path.
The Hibs goalkeeper then produced another decent block to deny a powerful effort from Sandaza as United went for broke.
Wilkie was also denied in the final stages when he saw his header palmed away.
Just when Hibs thought they had held on for the win, United claimed a share of the points in injury-time when Dixon's corner sparked a goalmouth scramble before substitute Goodwillie managed to bundle over the line.