There were also first-leg wins for Spanish duo Villarreal and Real Betis, plus Wisla Krakow, Rangers and Valerenga. A further 10 ties will be played on Wednesday.
The prize for the 16 winners is a place in the group stage of the Champions League, and the millions of pounds that come with it. The losers will have to settle for a slot in the UEFA Cup.
United, winners in 1999, are well used to playing in the group stage having done so 10 times previously, including a record run of nine in a row since 1996/97.
However, this is the second year running they have had to go through qualifying after finishing only third in the Premier League.
Visitors Debrecen arrived on the back of an 8-0 aggregate victory over Hajduk Split in the second round but they were never a factor at Old Trafford.
Wayne Rooney got United's season under way with a low shot after seven minutes, Ruud Van Nistelrooy scored the second four minutes into the second half and Cristiano Ronaldo capped an excellent individual performance with the third in the 63rd.
A reduced crowd of 51,701 watched the game, more than 15,000 short of capacity, with many fans boycotting the match in protest at the club's takeover by American tycoon Malcolm Glazer.
GOLD DUST
A few miles west tickets were like gold dust for Everton's return to the continent's premier competition after 34 years and their first European match since 1995.
However, the occasion ended in anti-climax for the locals with a 2-1 home defeat by Villarreal.
Everton were surprise fourth-placed finishers in the Premier League but found their spirited efforts not to be enough against a classy Villarreal side making their first appearance in the competition after finishing behind Real Madrid and Barcelona last season in Spain.
Superb goals by Argentina striker Luciano Figueroa (27) and midfielder Josico (45) either side of a close-range effort by James Beattie put the Spanish side, formidable at home, firmly in the driving seat.
Everton were twice denied the chance to play in the European Cup in the 1980s, due to the ban on English clubs after Heysel, and their last appearance was a defeat on away goals to Panathinaikos in the 1970-71 quarter-finals.
The Greek side have been regulars since then and got off to a flyer at Wisla Krakow with a fourth-minute goal by Nigeria-born Polish international striker Emmanuel Olisadebe.
IMPRESSIVE HOSTS
However, they ended up losing 3-1 after Pawel Brozyk, Kalu Uche and Tomasz Frankowski replied for the impressive hosts.
Real Betis left it late against Monaco after missing a host of chances in Seville as Edu struck in extra time to clinch a 1-0 win over the 2004 runners-up.
"I think the goal did us justice," said Betis coach Llorenc Serra Ferrer. "We dominated for most of the first half and the goal was a reward for our persistence.
Scottish champions Rangers, knocked out at this stage last season, look better positioned to go through this time after winning 2-1 in Cyprus against Anorthosis Famagusta with two goals in five minutes midway through the second half by Nacho Novo and Fernando Ricksen.
Valerenga will take a 1-0 lead into the second leg against Club Bruges after Steffen Iverson's 57th-minute effort gave them victory on the night in Oslo.
Wednesday's fixtures include holders Liverpool against CSKA Sofia, Udinese v Sporting Lisbon and Ukraine's Shakhtar Donetsk v Inter Milan.
The second legs of all 16 ties will be played on August 23 and 24 while the draw for the group stages is to be held in Monaco on August 25.
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