Wayne Rooney opened the way with a clinical finish after seven minutes and second half goals from Ruud van Nistelrooy and Cristiano Ronaldo followed.
The three sons of new United owner Malcolm Glazer -- Joel Avi and Bryan -- were at Old Trafford for the first competitive game under American ownership.
Several hundred fans demonstrated outside the 67500-seat stadium.
No incidents were reported but thousands of empty seats greeted the new owners.
On a visit in June shortly after Glazer took control of the club in his $1.47 billion takeover the three brothers needed a police escort from the stadium.
Debreceni had beaten Hajduk Split 8-0 in the previous round of qualifying and had been expected to give United yet to play a competitive game a solid test.
But they were a goal down before getting into their stride.
Rooney saw an attempted pass to Van Nistelrooy bounce back to him off a defender before firing a low shot into the keeper's bottom right hand corner for the opener.
Rooney and Ronaldo both fired wide of the target as United dominated the play in the opening half without moving out of first gear.
United started the second half as they had the first with an early goal.
Rooney fed by Darren Fletcher weighted his deep pass perfectly to give Van Nistelrooy time in front of goal and from 18 yards the Dutchman stroked the ball home for a 49th minute strike.
Van Nistelrooy then found Rooney darting into the penalty area with an intelligent 63rd minute pass and his astute cut back was rolled home by Ronaldo from close range.
Tamas Sandor clipped a free-kick over the wall but aimed too near to Edwin van der Sar as he wasted the visitors best chance.
United manager Alex Ferguson introduced Korean signing Park Ji-sung for his debut in the second half in front of a 51701 crowd more than 16000 below capacity.
Everton's return to European Cup action after a 34-year absence ended in disappointment when they lost 2-1 at home to Spain's Villarreal in the first leg of their third qualifying round tie.
Argentina striker Luciano Figueroa struck in the 27th minute in Villarreal's first serious attack driving in a superb low cross-shot.
Everton levelled three minutes before halftime when striker James Beattie pounced on a loose ball to poke it into the net from close range.
But the joy of the fans who had seen only two previous home European ties in the last 20 years was shortlived.
The visitors regained the lead in first-half injury time with a superb diving header by midfielder Josico to put them in a strong position to advance to the group stage in their first season in the competition.
The return leg is on August 24.