Maicosuel finished off an incisive passing move with a first-time shot from close range just after the half-hour mark to deny visiting Inter a likely derby tie with AC Milan in the last eight.
Massimiliano Allegri's Milan take on Serie B side Spezia at the San Siro on Wednesday.
Udinese's win is a boost for Francesco Guidolin's side who are 13th in Serie A having lost 10 of their 18 matches.
Inter, sixth in the league, dominated the tie but struggled to create chances.
Coach Rudi Garcia's Roma, chasing a 10th Italian Cup trophy, will face Juventus at the Stadio Olimpico later in the month after Vasileios Torosidis tapped the ball into the net for the hosts in the sixth minute.
They would have managed a more comfortable margin of victory but for a sterling display from visiting goalkeeper Vincenzo Fiorillo who among a series of saves pulled off an incredible double stop to deny Alessandro Florenzi.
The tie with Juve will be a chance for revenge for Roma after their 3-0 defeat by the Serie A champions on Sunday.