Claudio Ranieri's side, who lead Serie A by a point with four games left, went through 2-1 on aggregate thanks to February's 2-0 first-leg win.
Alexis Sanchez gave relegation-threatened Udinese a glimmer of hope nine minutes from time with a powerful shot from the right channel and the hosts could easily have levelled the tie late on when Bernardo Corradi headed straight at Julio Sergio.
Roma, who barely carved out a chance and had defender Marco Cassetti sent off for a second booking on 84 minutes, will meet title rivals Inter in the final in Rome on May 5 for the fifth time in six years.
Wednesday's defeat, coupled with losses to Panathinaikos in the Europa League last 32, constrast with a run of 24 games unbeaten in Serie A which has propelled the debt-laden club towards improbable glory.
"In the final minutes we were a little worried," Ranieri told reporters. "It was better that our good run ended today, let's hope our form in the championship lasts."
Ranieri rested Francesco Totti on the bench with one eye on the title run-in and following the captain's thumbs-down gesture towards Lazio fans after Sunday's volatile city derby which landed him a hefty fine.
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