For long spells, the match at Hamilton - which was initially claimed by the weather back on December 19 - did not appear to be worth the wait.
But Goodwillie netted the winner late on to rescue the meeting with Accies from being a drab goalless affair - and handed caretaker boss Peter Houston his first victory in the SPL.
It was no real surprise to see Hamilton name the same side who stunned Rangers in the Scottish Cup when they welcomed United to New Douglas Park for the rearranged SPL clash.
Accies fought back from behind to earn a 3-3 draw against the holders on Sunday and boss Billy Reid kept the faith with the same 11 players but this match failed to live up to the weekend's action in terms of entertainment.
Hamilton did look the more dangerous side early on and carved out a couple of chances in the opening spell.
Mickael Antoine-Curier's cross found Stuart Elliott who dispatched a bullet header towards goal. Garry Kenneth managed to block, amid claims he had used his hand, but the pleas for a penalty were ignored by referee Iain Brines.
James McArthur then slipped a lovely pass through for Marco Paixao but the Portuguese was not quite fast enough to beat goalkeeper Dusan Pernis to the ball.
At the other end, Goodwillie was not too far away with an angled drive that Tomas Cerny did well to palm to safety.
Hamilton still looked the more likely side to open the scoring and they squandered another opportunity when Alex Neil picked out Simon Mensing only for the midfielder to nod just wide of target.
Instead, it was United who almost snatched the opener. Mark McLaughlin conceded a free-kick for a foul on Goodwillie and Paul Dixon punted a dangerous ball into the box, with Neil's attempt at a clearance smacking the post.
A dreadful attempt at a backpass from Elliott then gifted Goodwillie possession, who bore down on goal and looked set to open the scoring just before the break only for Cerny to dive low and smother.
Elliott was replaced by James Wesolowski at half-time, before United were forced to swap the injured David Robertson for Danny Swanson just six minutes after the restart.
Once the action was under way again, Mensing whipped a cross in from the right for Antoine-Curier but he was denied by Pernis from close-range, with the offside flag raised anyway.
Both side were desperate for the goal that could seal the three points and McArthur resorted to a long-range effort that cannoned off a Tangerine jersey, before Antoine-Curier's shot from distance deflected just wide.
Paixao had the ball in the back of the net for Hamilton but was flagged for offside before the shot had left his boot.
Then, with 15 minutes to go, United also had the net bulging and there could be no complaints about this effort as Kenneth provided the cutback for Goodwillie to lash home from just inside the box, leaving Cerny helpless.
Accies tried to hit back for a late leveller through substitute Ali Crawford but his angled drive was well held by Pernis.
United could have had the chance to end any hope of an Accies comeback when McArthur appeared to foul Swanson in the box but, after consulting with his assistant, Brines opted not to award a penalty.