Substitute Harry Cornick curled Luton into an early second-half lead and when Stuart Dallas fired Leeds level just after the hour, it appeared there would be only one winner.
But Luton, who climbed off the bottom of the table after Saturday’s win at Swansea, withstood a second-half onslaught and gave boss Nathan Jones everything in their fight for survival – but despite the point, Jones’ men are still favourites for relegation with the Championship odds from Paddy Power.
Unbeaten in their last five matches, the Hatters were confident, well-drilled and full of running, while goalkeeper Simon Sluga made several excellent saves.
Patrick Bamford forced Sluga into a near-post block before Leeds were dealt an early blow when skipper Liam Cooper was forced out of the action.
The centre-half failed to recover after taking a knock as he defended a corner and was replaced by Gaetano Berardi in the 12th minute.
Luton’s appeals for a penalty were waved away by referee John Brooks after Berardi had tangled with Callum McManaman, but it was a sprightly start from the visitors.
Helda Costa was also off target for Leeds, while Luton served a warning of their threat from set-pieces when captain Sonny Bradley’s towering header was saved by Illan Meslier.
Leeds forced a succession of corners which came to nothing as half-time approached and Luton boss Nathan Jones will have been the happier manager during the interval.
Bielsa withdrew Berardi for Ezgjan Alioski and Jones replaced McManaman with Cornick, who wasted little time in curling Luton into a 50th-minute lead.
Ryan Tunnicliffe picked him out with a raking upfield pass and Cornick went one-on-one with Ben White before finishing with aplomb.
Leeds’ response was immediate, but it found Sluga in inspired form as the Croatian goalkeeper pulled off fine saves to keep out Roberts’ deflected shot and then successive efforts from Patrick Bamford and Alioski.
Roberts then crashed a shot just wide, but two minutes after Pablo Hernandez had replaced Mateusz Klich, Leeds drew level.
Alioski found space on the left and his ball into the area found Dallas, who had time enough to pick out the bottom corner with a neat low finish.
Bamford’s first-time effort following another excellent ball in from Alioski was narrowly wide as Leeds chased a second goal.
Costa thumped a low shot past the post and then just failed to make full contact from Alioski’s cross from in front of goal as gutsy Luton held firm.