Rampant Roma snatch semi-final advantage
Mirko Vucinic latched on to a pass from deep by Daniele De Rossi before slotting the ball into the corner and Philippe Mexes headed home in the first half to put the hosts in command of the tie.
The victory stretched Roma's unbeaten run in all competitions to 18 games, having moved level on points with second-placed AC Milan in Serie A on Sunday with a 2-1 win over Siena.
On Thursday, they played some neat one-touch football in the opening phase with captain Francesco Totti back from injury along with fellow striker Vucinic, who gave them a 12th-minute lead with the help of some distracted defending.
Udinese, who are fourth-from-bottom in Serie A, had more possession as the first half wore on but created few chances.
French defender Mexes beat their back-line to a David Pizarro free-kick to extend Roma's lead in the 40th.
The home side cruised in the second half and Udinese's failure to make an impression on the score-sheet leaves them facing an uphill task in the return leg, which is scheduled to take place in April.
Serie A leaders Inter Milan beat Fiorentina 1-0 at the San Siro on Wednesday in the first leg of the other semi.