Goodwillie was missing after United agreed a fee with Blackburn for the 22-year-old striker, but the visitors put the potential departure of their talisman behind them when Jon Daly nodded in Willo Flood's 38th-minute corner.
In a riveting contest between the teams who finished third and fourth in the SPL last term, John Rankin had earlier hit the bar with a free-kick for United, while Hearts had a number of opportunities themselves, with Stephen Elliott prodding wide after a neat turn soon after the hosts had gone behind.
David Templeton was denied by goalkeeper Dusan Pernis in the second half as United showed they are more than a one-man team and there will be life after Goodwillie.
Hearts perhaps paid the price for their Europa League qualifying excursion to Hungary, where they drew 1-1 with Paks, with manager Jim Jefferies opting for a solitary enforced change.
The hosts started well and a Marius Zaliukas shot ricocheted off Elliott, who started in place of the suspended Jamie Hamill, and narrowly wide, before a Ryan McGowan long throw had to be flapped away by Pernis.
The visitors had a good opportunity when they were awarded a free-kick after Adrian Mrowiec was adjudged to have impeded Rankin on the edge of the box.
Referee Brian Winter immediately blew for the foul, but insisted it was outside the area despite the former Hibernian midfielder tumbling inside.
Rankin struck the subsequent free-kick which beat Hearts goalkeeper Marian Kello but came back off the bar.
Pernis was soon called into action again at the other end, saving Elliott's effort from Danny Grainger's cross before John Sutton diverted another cross from the left narrowly wide.
Kello was at full stretch to deny Daly as the United captain met Paul Dixon's left-wing cross with a first-time volley, but from the subsequent corner the Hearts goalkeeper was beaten.
Flood's set-piece found Daly unmarked eight yards out and the striker headed United in front.
Elliott had another opportunity after controlling Eggert Jonsson's drive before prodding off target when he perhaps should have done better as Hearts entered the interval behind.
The hosts began the second half well as Grainger cut in off the left and shaped to shoot, only to be brought down by Flood.
The United midfielder earned the game's first booking for the foul but the visitors escaped further punishment as Templeton curled the free-kick wide.
Kello then saved from Scott Allan, making his first United start, after Hearts only half-cleared a Flood corner before the Hearts stopper held a Garry Kenneth header.
United were causing Hearts trouble and Johnny Russell twisted before drilling a left-foot shot across goal and wide of the upright.
Hearts soon had Pernis in action again as Templeton latched onto Sutton's flick-on from Kello's long ball forward before hitting a blistering drive which the United number one turned behind for a corner.
Jefferies introduced the fit again Rudi Skacel and Arvydas Novikovas for Elliott and Mrowiec with 20 minutes remaining, seeking a fresh impetus.
United replaced Scott Severin with Scott Robertson and Hearts pulled off Sutton for Ryan Stevenson as the changes continued before Grainger was booked following a challenge on Flood.
Hearts kept the pressure on, but struggled to create chances.
Kenneth recovered from a lapse in concentration to intercept a ball bound for Skacel, and Stevenson saw a hooked effort from a narrow angle go along the goalline as the United defence stood firm and the visitors celebrated victory.