Marko Arnautović and Kristjan Asllani netted in the first half to help Inter set up a last-eight clash against Lazio - the team they thrashed 6-0 in the league on Monday.
Inter coach Simone Inzaghi made eight changes to that side.
The match was briefly halted shortly before half-time when a spectator collapsed in the stands. The fan was immediately treated as the stadium fell into silence and the players looked on, clearly concerned.
A defibrillator had to be used before the fan was carried out on a stretcher to the applause of the San Siro crowd. The fan was reportedly stable in the hospital.
When play resumed after a delay of more than five minutes, the Udinese players still seemed distracted as Asllani's corner from the left evaded everyone and went in off the far post.
That put Inter 2-0 up as they had broken the deadlock in the 30th minute following an Udinese error.
A hideous pass from visiting midfielder Jurgen Ekkelenkamp went straight to Mehdi Taremi and he played in Arnautović, who slotted into the bottom right corner.
Taremi hit the post in the second half, while Inter also had an early penalty revoked on review.
Teenage defender Mike Aidoo came on two minutes from time for Inter for his professional debut.
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