Udinese, struggling in Serie A, booked a semi-final spot with AS Roma next month when Swiss midfielder Inler slotted home 11 minutes into the second half after Milan left back Luca Antonini was dispossessed in his own box.
A second-string Milan were under the cosh from the kick-off and barely mustered a decent effort on goal with Dutch striker Klaas-Jan Huntelaar, linked in the media with a transfer, particularly ineffective.
The only bright spot for the Rossoneri, beaten 2-0 by Inter Milan in Serie A on Sunday, was the long-awaited return from injury of goalkeeper Christian Abbiati.
"We struggled to increase the pace. We didn't push enough to get that goal," Milan coach Leonardo told Rai television.
Inter host Juventus in the final last-eight tie on the same battered San Siro pitch on Thursday in what could be Juve coach Ciro Ferrara's last game in charge amid media speculation he will be sacked.
Fiorentina will face the winners of Thursday's match after beating holders Lazio 3-2 last week in the much-maligned competition.
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