Craig Brown's dream start as Aberdeen manager looked set to continue when a lapse in concentration from Keith Watson gifted the Dons the opener.
Chris Maguire raced on to the full-back's short back-pass before rounding Dusan Pernis and slotting home after 37 minutes.
The Dons dominated the second period but could not find a second goal to kill off a below-par United and Goodwillie popped up in the fifth minute of added on time to volley past Jamie Langfield and earn his side a share of the spoils.
Full-back David McNamee was handed his Dons debut in place of Nikola Vujadinovic, who picked up an injury in the 1-0 win over Hamilton Accies on Wednesday.
Winger Sone Aluko and youngster Ryan Jack also started with Derek Young and Josh Magennis dropping to the bench.
United, without the suspended Danny Swanson following his red card in their 2-2 draw with Hibernian, made only one change with Garry Kenneth taking his place at the heart of defence after returning from a ban.
Jamie Langfield prevented Aberdeen conceding an early goal when he pulled off an instinctive stop to keep out Goodwillie's close-range volley after four minutes.
The game continued at a frantic pace with Aberdeen surging up the other end, spearheaded by Aluko but the Nigeria international delayed his pass to Maguire after an enterprising run allowing Sean Dillon to make a vital block.
Dillon picked up the first caution of the game after 18 minutes when he brought down Maguire as he raced down the right flank before Morgaro Gomis joined him in the book for a late challenge on Jack.
Scott Vernon, who bagged his eighth goal of the season with a late winner against Accies midweek, had a chance to break the deadlock but the former Colchester striker headed over from a Paul Hartley free-kick.
United created the next opening after 31 minutes when Goodwillie managed to out-muscle Zander Diamond and crossed for the inrushing Prince Buaben but the Ghanaian could not divert his downward header past Langfield.
The visitors had been enjoying a period of dominance but it was Aberdeen who opened the scoring after 37 minutes courtesy of a careless error by Watson.
The right-back's pass was woefully short and Maguire immediately latched on to the loose ball and rounded Pernis before passing the ball into the unguarded net.
United made an attack-minded switch at the break with striker Daly introduced at the expense of Senegalese midfielder Gomis.
But it was the Dons who continued to press for a second with an appeal for a penalty waved away after Maguire fell in the box under pressure from Paul Dixon before Vernon flashed a low shot wide of the target.
The Dons had United pinned back in their own half and were creating the better of the chances but were forced to make a change after 66 minutes when Yoann Folly was stretchered off with a serious-looking injury following a challenge from Dixon and was replaced by Young.
With 15 minutes remaining both sides made changes with Josh Magennis replacing Jack for the Dons, while United introduced David Robertson and Stuart Armstrong for Scott Robertson and Barry Douglas.
McNamee's solid debut came to a premature end as he limped off with cramp and was substituted for Jerel Ifil.
The home side's relentless pursuit of a second continued with Maguire unlucky to see his well-struck shot blocked by Dillon.
And the Dons paid the price for their profligacy when, in the fifth of six minutes of injury time, Dillon's cross into the box was headed down by David Robertson for Goodwillie who volleyed the ball past Langfield and silence the home support.