The result has left the Blues in fourth spot, three points ahead of fifth-placed West Ham United, who now have two games in hand over their capital rivals, while Leeds remain in 11th position.
Both teams had opportunities to make the breakthrough in the first half, with Chelsea coming close through Kai Havertz and Ben Chilwell, while the Blues also hit the Leeds crossbar when the ball bounced off Diego Llorente.
Tyler Roberts had the home side's best chance of the first period when he attempted to pick out the top corner in the 16th minute, but Chelsea goalkeeper Edouard Mendy tipped the attacker's effort onto the crossbar.
Thomas Tuchel's team had more of the ball in the opening 45 minutes, but the half-time whistle arrived with the two sides level at 0-0, leaving both managers with plenty to ponder.
Chelsea continued to dominate as the second period progressed, but opportunities were few and far between in the league affair as the points were ultimately shared.
Next up for the Blues is the second leg of their last-16 Champions League clash with Atletico Madrid on March 17 before facing Sheffield United in the quarter-finals of the FA Cup four days later. Leeds, meanwhile, will continue their Premier League campaign away to Fulham on March 19.
7th min: Huge chance for Chelsea to make the breakthrough as Christian Pulisic delivers a low cross into Havertz, but the Germany international cannot make proper contact on the ball.
7th min: Leeds go straight down the other end and have the ball in the back of the Chelsea net as Roberts converts a low Patrick Bamford cross from close range, but the offside flag is raised, with Bamford just going too early.
10th min: Chelsea come close again as Luke Ayling smashes the ball at Leeds teammate Llorente and it drops over the head of Illan Meslier, but the crossbar comes to the home side's rescue and Meslier is on hand to collect the rebound.
We NEARLY witnessed a very bizarre own-goal here! 😳— Football on BT Sport (@btsportfootball) March 13, 2021
Ayling clears the ball straight into Llorente and the ball ricochets off the bar!
So close to an opener... pic.twitter.com/Dgb1jg2xah
15th min: Another opportunity for the visitors as Pulisic delivers a low cross into the Leeds box, but Meslier does well to claw it away before Ayling clears his lines.
16th min: Roberts hits the crossbar for Leeds as he attempts to pick out the far corner, and replays show that Chelsea goalkeeper Mendy just managed to get his fingertips on the ball.
A frantic start at Elland Road! 🔥— Football on BT Sport (@btsportfootball) March 13, 2021
Tyler Roberts hits a glorious curling effort, which Edouard Mendy just gets his fingertips too! 🧤
That would have been some goal... pic.twitter.com/ma9jGhdary
33rd min: Chelsea work the ball impressively in a forward position before Chilwell strikes towards the bottom corner, but the left-back's effort is deflected just wide of the Leeds post.
38th min: Antonio Rudiger finds himself free inside the Leeds box as a corner drops to the centre-back, but the ball drops harmlessly into the arms of the home side's goalkeeper.
46th min: Half-chance for the Blues just before the interval as Hakim Ziyech's pass releases Havertz, but the German's effort is straight at the Leeds stopper.
48th min: Big opportunity for Chelsea to take the lead early in the second half as Havertz breaks into the Leeds box, but Meslier is on hand to make a smart save and tip the ball over for a corner.
53rd min: Ziyech delivers a wonderful cross into the Leeds box, Havertz is again on the scene, but the summer arrival again fails to make proper contact on the ball.
55th min: Huge save from Mendy to keep out a shot from Raphinha after the Leeds attacker had spun inside the penalty box and managed to fire an effort towards goal.
62nd min: Chilwell drives into the Leeds box and releases a shot, but the defender is under pressure from Jack Harrison and can only fire wide of the post.
72nd min: Meslier has to get down to keep out a long-range strike from Rudiger as Chelsea continue to search for the opening goal of the match at Elland Road.
77th min: Leeds come close as Llorente strikes one from outside the box, but Reece James just manages to deflect the Spaniard's effort over the crossbar.
78th min: Free header for Rodrigo, but the Leeds attacker heads straight at Mendy.
87th min: Callum Hudson-Odoi works some space for the visitors before dropping his shoulder and attempting to pick out the far corner, but the Leeds goalkeeper makes a smart save.
MAN OF THE MATCH - KALVIN PHILLIPS
Kalvin Phillips was a key contributor to Leeds picking up a point at Elland Road, with the England international enjoying an excellent match in the middle of the park.
The 25-year-old had a pass success rate of 91%, while he also made four tackles, and the midfielder's energy was key for Bielsa's side, particularly in the second period as Chelsea dominated the ball.
Chelsea goalkeeper Mendy was in excellent form on Saturday, making two brilliant saves, the first of which was a stunning stop to prevent a clever effort from Roberts from finding the back of the net in the first period.
The summer arrival also kept a close-range strike from Raphinha in the second half as the 29-year-old played a leading role in Tuchel's side keeping another clean sheet.
14% - Of the 48 opponents Chelsea have faced away from home at least 15 times in all competitions, the Blues' lowest win rate is against Leeds (14% - P49 W7 D14 L28). Foes pic.twitter.com/1LKPpAYsAW— OptaJoe (@OptaJoe) March 13, 2021
Possession: Leeds 38%-62% Chelsea
Shots: Leeds 7-15 Chelsea
Shots on target: Leeds 4-8 Chelsea
Corners: Leeds 8-5 Chelsea
Fouls: Leeds 16-6 Chelsea
LEEDS (4-1-4-1): Meslier; Ayling, Llorente, Struijk, Alioski; Phillips; Raphinha, Roberts, Dallas, Harrison (Costa 64'); Bamford (Rodrigo 35', Klich 79')
CHELSEA (4-2-3-1): Mendy; Azpilicueta, Christensen, Rudiger, Chilwell; Kante, Jorginho; Ziyech (Werner 69'), Mount (Hudson-Odoi 79'), Pulisic (James 68'); Havertz