Mauro Icardi had given Inter the lead before half-time at San Siro, but Bruno Fernandes' second-half volley restored parity.
Inter were ultimately the architects of their own fate as Rodrigo Palacio's poor backpass allowed Cyril Thereau in on goal for Udinese's winner, leaving the hosts awaiting a win domestically since Mancini returned as manager.
Mancini was angry at the way his side allowed Udinese back into the match, but remained keen to point out plenty of positives to build on in the coming weeks.
"I am angry at this defeat," he told Sky Sport Italia. "We cannot do anything now, the guys are working well, we just have to roll our sleeves up and continue.
"In the first half we played well, we had so many chances and could have scored more.
"In the second half we started well, but then there was 15 minutes when Udinese took advantage."
For Udinese boss Andrea Stramaccioni, it was a big victory on his first return to Inter since being sacked in 2013, and he hailed a deserved three points.
"It was important for us to win," he said. "I think overall we deserved it, especially for the second half.
"We wanted to press high and we did not succeed [in the first half], but we reorganised and went more aggressive, these are three valuable points from which to build."
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