The Reds were second best for large parts of the second half, coming close to falling behind when Kemar Roofe struck the post, but they dug deep for the second home match running to secure the win late on.
It was a particularly special occasion for Ben Woodburn, who came off the bench to add to Divock Origi's key breakthrough goal, extending Liverpool's unbeaten run to 16 games and becoming the club's youngest-ever scorer in the process.
Leeds had a glorious chance to make the breakthrough just three minutes into the tie, only for Simon Mignolet to keep out Hadi Sacko after the winger latched on to a Stuart Dallas ball through the middle.
That would prove to be the only real intricate piece of play of the opening 45 minutes from either side, however, with Liverpool - showing eight changes from last time out - dominating possession but doing very little with it.
Roofe was the next visiting player to test Mignolet, this time trying his luck from 20 yards out with an attempt that the Belgian was always favourite to keep out.
After spending almost the entire of the opening quarter of the game inside their own half, United grew more and more into things and looked just about the more threatening as things transpired.
For all their passing moves, it was a slack piece of Leeds play which gave Liverpool their only other sight of goal before the break - Emre Can making the most of some hesitancy at the back and prodding the ball marginally wide.
That provided the Premier League title challengers with a stark warning that progression was not guaranteed - a point that was hammered home 10 minutes into the second period when, after latching on to a stray Kevin Stewart pass, Roofe sent a curled attempt back off the upright.
Can rather reluctantly fired in a shot at the other end soon after, which took a deflection and ended safely in the hands of Silvestri, but Garry Monk was growing more and more confident and turned to Chris Wood - the competition's second-top scorer - on the hour.
That attacking move almost paid off six minutes later when a disjointed Liverpool attack broke down, allowing Ronaldo Vieira to play in Roofe who was this time kept out by Mignolet down low.
Jurgen Klopp's men finally woke from their slumber 20 minutes from time, with Wijnaldum placing the ball across Silvestri but back off the post, and neither Sadio Mane nor substitute Woodburn could convert from the rebound.
The opening goal finally arrived for Liverpool in the final quarter of the match thanks to a poacher's finish from Origi, who gambled by getting on the end of Trent Alexander-Arnold's right-sided cross to prod the ball past Silvestri near his front post.
Liverpool did not have to wait too long for the second to arrive, either, and it was a moment to remember for Woodburn who was left with the task of rifling the ball into the roof of the net when the ball was worked to his feet inside the box.
Leeds, who have now lost 10 of their last 12 games against top-flight opposition in this competition, offered a good account of themselves late on - Wood seeing a goal ruled out for offside and Sacko failing to convert when again played through on a night that could have ended much differently for the visitors.