Albanian striker Erjon Bogdani's 31st minute header and a superb finish from fellow forward Emanuele Giaccherini just before the break left visiting Milan in complete shock and rueing two Alexandre Pato strikes which were ruled out.
Former Barcelona and Inter man Ibrahimovic missed a late penalty as predictions the Swede could swing the Serie A championship race Milan's way this term after five straight Inter titles began to look premature.
Robinho, Milan's other big transfer window signing, came on for Ronaldinho in the second period but the Brazilian could not silence the capacity 24,000 crowd in the small city near Rimini.
The score was already 1-1 at that point following goals from Cagliari's Daniele Conti, whose father Bruno is Roma sporting director, and a Daniele De Rossi header.
Alessandro Matri, who netted the earlier spotkick after ex-Inter man Burdisso's nasty tackle on the then substituted Conti, scored again before Andrea Lazzari's late fifth.
Lucio bundled in Wesley Sneijder's corner following a mistake from goalkeeper Samir Handanovic early in the first half before Antonio Floro Flores headed an equaliser.
Handanovic then blocked Samuel Eto'o's second-half spotkick after Gabrielle Angella's handball but the Cameroon forward buried the rebound before also hitting the inside of the post.
"We know his quality, it's only a matter of time," new coach Rafael Benitez told Sky. "We started well but then lost control of the midfield and they had some dangerous counter attacks."
