The on-loan striker from Blackburn took advantage of a slip by Thistle defender Conrad Balatoni in the 33rd minute to open the scoring with a shot from 10 yards.
Just before the interval Goodwillie grabbed a second with a header from a Mark Wilson corner to stun the visitors who went into the dressing room wondering how they were two down.
Partick came out for the start of the second-half determined to claw their way back into the cup-tie.
The United defence found themselves having to defend for more than they or their fans in the sparse Tannadice crowd would have liked for a period before they asserted themselves to find a third in the 85th minute through substitute Ryan Dow, on for Brian Graham.
Christie Elliott scored a consolation for the visitors in the final minute but Goodwillie was not to be denied the match ball by scoring from a Stuart Armstrong pass as United ran out comfortable winners in the end.
With Celtic knocked out by Championship side Morton on Tuesday night, the Tayside club go into the quarter-final draw at Hampden on Thursday afternoon with as much chance as anyone of going all the way in the competition.
- Sports & Recreation
- Conrad Balatoni