Inler struck in the 56th minute to hand Udinese their first away road win of the season and set up an encounter with Roma in the final four next month.
Milan coach Leonardo made wholesale changes to the team that lost 2-0 at Inter Milan on Sunday, with Klaas-Jan Huntelaar paired alongside Filippo Inzaghi in attack.
But both strikers failed to test Udinese goalkeeper Samir Handanovic on the night as the visitors ran out comfortable winners.
Udinese dominated the encounter at the San Siro and were unlucky not to take a 25th-minute lead when Alexis Sanchez struck the post with a right-footed strike from the edge of the area.
The hosts had their first chance in the 28th minute with Marek Jankulovski's 30-yard strike forcing Handanovic to clear.
Handanovic was called on again shortly afterwards to save Mathieu Flamini's long-range effort.
Udinese almost took the lead on the half-hour mark but Milan goalkeeper Christian Abbiati did well to palm over Antonio Di Natale's right-footed strike.
Sanchez should have broken the deadlock before half-time but he failed to beat Abbiati in a one-on-one.
After the restart, Udinese picked up where they left off and their efforts were soon rewarded.
Sanchez stole the ball from Luca Antonini and played in Inler, whose powerful right-footed strike from the edge of the area beat Abbiati.
Marco Borriello replaced Huntelaar on the hour mark but the Rossoneri struggled to break down Udinese's defence.
In the 66th minute, Gianluca Zambrotta almost struck the equaliser, the Italy international surging towards the area but skewing his left-footed strike went wide.
With 15 minutes remaining, Floro Flores should have killed the game off but his lob went over the bar.
Udinese held on for a deserved victory as Milan bowed out of a competition they have not won since the 2002-03 campaign.
"It's disappointing," said Leonardo. "We wanted to go far in the competition, but we should have given more rhythm to our play and we didn't do that.
"We lacked the right movements in attack and we weren't dangerous. The only positive aspect was the performance of Abbiati tonight on his return from injury."