The former Huddersfield striker netted a fourth-minute penalty and secured the points with a late counter-attack.
Michael Stewart was sent off in his first match since being handed the Hearts captaincy as Csaba Laszlo's side suffered a blow ahead of their Europa League play-off against Dinamo Zagreb.
The 52nd-minute red card - Hearts' fifth in four visits to Tannadice - nullified the Edinburgh side's comeback attempts as they again paid the price for their poor disciplinary record.
They were already missing vice-captain Marius Zaliukas, who is serving a four-match ban following his four red cards last season.
Laszlo handed winger Suso Santana and striker David Witteveen debuts in a positive formation but they were unable to breach a United defence which featured encouraging introductions for goalkeeper Nicky Weaver and on-loan Rangers defender Andy Webster.
There was a new look to the Hearts defence and it showed enough vulnerability to cause concern ahead of Thursday's clash in Croatia.
Right-back Eggert Jonsson gifted the penalty when he clumsily tripped Danny Swanson just inside the box after the United midfielder had cut past him.
And Ismael Bouzid, who partnered fellow debutant Dawid Kucharski in the centre with Jose Goncalves at left-back following his year-long loan spell at Nuremberg, was fortunate not to concede another.
They were tested by the running of David Goodwillie and Cadamarteri, who slotted the penalty into the corner despite Marian Kello getting a hand to the ball.
The early goal saw United sit deeper but their lead had not been troubled before Bouzid's sliding challenge felled Cadamarteri as he ran on to Goodwillie's flick in the 18th minute. The Algeria defender took nothing of the ball but Steve Conroy waved play on.
Weaver made his first save midway through the half when he dived to catch a long-range effort from Ruben Palazuelos. Witteveen then headed over under pressure from Webster, who was roundly booed by the visiting fans as he made only his 12th club appearance since buying his way out of his Hearts contract three years ago.
Most of the Hearts threat was coming from set-pieces and Goncalves was only inches away from connecting with David Obua's curling free-kick into the goalmouth.
Witteveen evaded Webster to pounce on Obua's knock-down following another free-kick but Weaver was off his line quickly to smother and the former Manchester City keeper dived low to push away Obua's effort just before the break.
Goodwillie shot high and wide after beating Bouzid to Paul Dixon's long ball but Hearts were soon at a further disadvantage when Stewart, who had been booked for a first-half foul on Swanson, earned a second yellow card when he slid in late on Scott Robertson.
Laszlo brought on Ian Black for his debut in place of Witteveen and his bad tackle on Swanson sparked a confrontation that saw both players and Santana booked.
Kello brilliantly turned over Goodwillie's controlled volley following a cross from substitute Craig Conway, and Webster headed against the outside of the post following the winger's deep corner.
Although Hearts saw a decent share of possession, they struggled to create chances and Cadamarteri finished off the contest in the 85th minute.
Prince Buaben sent him clear from deep inside his half after United had cleared a free-kick and Cadamarteri slid the ball past Kello, who had raced well outside his box to try to close down the Englishman. Weaver further frustrated the visitors when he pushed over Jonsson's header.
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