Jack Harrison opened the scoring with a stunning strike from outside the penalty box, before Junior Firpo and Luis Sinisterra both found the net midway through the second half to put the Whites in control.
Accrington supporters had something to cheer in the final 10 minutes when 18-year-old Leslie Adekoya pulled one back, but the Premier League side ultimately did enough to see off their spirited League One counterparts.
The 1972 FA Cup winners will now await their fifth-round opponents when the draw is made at 7.00pm on Monday.
In front of around 5,000 fans at the Wham Stadium, John Coleman's side started the brighter of the two sides, striking six attempts on goal in the first 15 minutes, but the best chance in the opening exchanges fell to Leeds debutant and club-record signing Georginio Rutter, who fizzed a left-footed shot narrowly wide of the post.
Leeds gradually grew into the contest and eventually broke the deadlock in the 23rd minute courtesy of a moment of real Premier League quality from Harrison.
After receiving a lay-off from Patrick Bamford, Harrison drilled a first-time strike from around 25 yards out into the bottom-right corner, with the pace and power of the shot beating goalkeeper Toby Savin all ends up.
Sinisterra almost doubled Leeds' lead with a speculative effort in the 35th minute, but his right-footed strike was brilliantly saved by the fingertips of Savin before it cannoned off the woodwork.
Liam Coyle received the first yellow card of the game six minutes before the break for a strong challenge of Firpo, which sparked an altercations between Leeds boss Jesse Marsch and Accrington assistant Jimmy Bell on the touchline.
Shortly after, Rasmus Kristensen drilled a low curling shot across the face of goal and narrowly wide of the far post before a deflected effort from Ethan Hamilton was comfortable collected by Illan Meslier.
Accrington came out the blocks quickly in the second half and almost netted an equaliser less than a minute after the interval, but Harvey Rodgers somehow directed his effort from close range over the crossbar.
Leeds responded well, though, and were driven on by the energetic support from their traveling away fans as they began to gain a foothold in the game.
The visitors ultimately capitalised on Accrington's tired legs and put the tie to bed midway through the second half with a quickfire double from Firpo and Sinisterra.
A superb team move by Leeds, including a skillful swivelled through-ball from Bamford, was finished off by Firpo in the 66th minute, with the left-back slotted home his first-ever Leeds goal.
Less than two minutes later, Harrison drove into the danger zone from the left flank before setting up Sinisterra for a simple finish from around eight yards out.
Accrington then pulled one back 10 minutes from time through teenage substitute Adekoya, who placed a right-footed finish over Meslier with his first touch just 38 seconds after coming off the bench, becoming the youngest player to score in this season's FA Cup.
Adekoya came close to doubling his tally with a lofted header over the crossbar in stoppage time, but Leeds held on to book their place in the fifth round.