Both sides were denied by the crossbar as Tyler Roberts came closest for Leeds in a game in which United lost Patrick Bamford to injury in the first half.
Leeds had gone 18 league games without drawing since the clash against Arsenal at Elland Road on November 22 which also ended 0-0.
Chelsea went close to taking a seventh-minute lead when Kai Havertz latched on to a Christian Pulisic cross but his weak attempt was saved by Illan Meslier.
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Leeds immediately countered and Tyler Roberts netted a tap in from Bamford's square ball 17 seconds later but Bamford had strayed offside.
The Whites then had a very narrow escape in the tenth minute when a Luke Ayling clearance smacked back off Diego Llorente and on to the crossbar before Meslier caught the rebound.
But Leeds were then denied by the woodwork in the 15th minute as a lovely dinked effort from Roberts was tipped on to the bar by Edouard Mendy.
United then lost Bamford to injury as Rodrigo replaced the Whites no 9 from the bench in the 36th minute and Chelsea finished the half with efforts form Antonio Rudiger, Mason Mount and Havertz who all failed to find the back of the net.
Havertz was presented with another fine chance three minutes after the restart but saw his rising drive parried over by Meslier.
The German then failed to beat Meslier with a close range header as Chelsea turned the screw.
But only a fine save from Mendy denied Leeds a 55th-minute opener as the Chelsea 'keeper tipped Raphinha's effort from close range behind for a corner.
Pulisic's effort from a tight angle was also saved by Meslier who later tipped a Rudiger drive away but Llorente's effort for Leeds from a fine Roberts pull back was deflected over the bar.
Rodrigo then put a free header from a Raphinha corner straight at Mendy and was taken off for Mateusz Klich shortly after.
As the game fizzled out, Blues substitute Callum Hudson-Odoi fired straight at Meslier and it proved the last real chance as Leeds survived a late Chelsea corner and both sides took a point.
Leeds United: Meslier, Ayling, Llorente, Struijk, Alioski, Phillips, Dallas, Harrison (Costa 64), Raphinha, Roberts, Bamford (Rodrigo 36, Klich 79). Subs not used: Casilla, Koch, Berardi, Shackleton, Jenkins, Poveda.
Chelsea: Mendy, Azpilicueta, Rudiger, Christensen, Chilwell, Ziyech (Werner 69), Jorginho, Kante, Mount (Hudson-Odoi 79), Havertz, Pulisic (James 69). Subs not used: Kepa, Alonso, Zouma, Emerson, Kovacic, Giroud.
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Thank you Laura Collins