Hungarian champions Debrecen took an early lead through Peter Czvitkovics's tap in from close range but Mutu volleyed in Dario Dainelli's cross and then set up Alberto Gilardino to fire the Italian side ahead after just 10 minutes.
In a first-half goal blitz, Mutu blasted in his second after Gilardino returned the favour to his strike partner after escaping his marker at Ferenc Puskas stadium.
"Mutu produced a top-draw performance which obviously helps the team to play at a higher level. I knew he was going to do something extra after his approach and work in recent weeks," said Fiorentina coach Cesare Prandelli.
Gergely Rudolf pulled one back for the hosts with a superb strike, curling a shot into the top left corner after teasing two defenders with his trickery. Mario Santana then restored Fiorentina's two-goal cushion eight minutes before halftime.
The second-half was a quieter affair with French striker Adamo Coulibaly heading a third goal for Debrecen two minutes from time but the Hungarian side are still waiting for their first Champions League point.
"If someone had told me before the game we would score three goals and not get a point I would not have believed it," Debrecen coach Andras Herczeg told reporters.
Olympique Lyon have a maximum nine points with Fiorentina on six, Liverpool three and Debrecen yet to register a point after three games.
