The two sides went head-to-head in this competition for the first time since their famous 1973 final, which Sunderland edged against the odds 1-0, and Leeds rarely looked like getting a measure of revenge at the Stadium of Light.
Patrick van Aanholt's first-half strike proved decisive to give Gus Poyet, who spent 12 months as Leeds assistant manager in 2006-07, a win over his former club.
Sunderland boss Poyet named a strong line-up, including the likes of Steven Fletcher, John O'Shea and Emanuele Giaccherini.
And, after seeing several excellent opportunities spurned inside the first half-an-hour, the hosts eventually opened the scoring in the 33rd minute when Van Aanholt rifled a shot into the bottom-right corner.
Leeds produced some improvement at the start of the second period, but the Championship relegation candidates were unable to make the most of their opportunities, with Liam Cooper hitting the post with a header late on.
United initially looked to take the game to their hosts, though the Premier League side's superior quality quickly became evident after a frantic few moments in the 12th minute.
First, Giaccherini cut inside an unconvincing challenge from Gaetano Berardi and forced Marco Silvestri into a save, before Ricky Alvarez struck the crossbar from the resulting corner.
Sunderland continued to put Leeds under significant pressure and went close for a third time in the 21st minute, but Fletcher failed to keep his effort down from six yards following Van Aanholt's cross.
Leeds had another escape five minutes later as Casper Sloth cleared Santiago Vergini's goal-bound header off the line.
Sunderland finally broke the Championship side's resistance just after the half-hour mark. Van Aanholt latched on to Fletcher's pass and drilled a first-time effort across goal from the left, with his fierce effort finding the net via the far post.
Leeds began the second half far better and went close to finding the net twice inside three minutes as goalkeeper Costel Pantilimon was called into action by Adryan and Brian Montenegro.
At the other end, Italian shot-stopper Silvestri once again denied his compatriot Giaccherini, as the midfielder looked for the bottom-left corner from 20 yards.
Silvestri continued his impressive display with 22 minutes to go, launching himself across goal to keep out an effort from Fletcher that looked destined to find the net.
Leeds committed men forward in the closing stages and they almost found the last-gasp leveller they were searching for as Cooper met Luke Murphy's cross with a fine header in the second minute of stoppage time, but Sunderland were saved by the left-hand post.
