Goodwillie, on loan from Blackburn in his second spell on Tayside, slotted home twice in the first half before securing his treble on the break in stoppage-time.
The visitors began the brightest, though, with Ross Forbes going close with a long-range shot as they showed their intent.
Brian Graham then went close for United only to be denied by Scott Fox but the Thistle keeper could do nothing to prevent Goodwillie opening his account for the night.
The Scotland internationalist took advantage of a slip from defender Conrad Balatoni to score from 10 yards after 33 minutes.
His second came before the break as he headed home a Marc Wilson corner to stun the Jags, who had battled hard with little reward.
Partick tried to get back into the contest in the second half but they struggled to create clearcut opportunities.
United substitute Ryan Dow put the game beyond doubt five minutes from time after Goodwillie had seen a shot come back off the upright.
The away side finally got some reward for their endeavours when Christie Elliott slotted home.
But Alan Archibald's side were punished as they pushed forward for another, with Goodwillie wrapping up his treble with a neat finish on the break after Stuart Armstrong had played him in.