Richard Cresswell hit the winner to throw Leeds a survival lifeline as fellow strugglers Luton were left to rue a late penalty miss at Elland Road.
Jonathan Douglas headed onto the bar twice and Robbie Blake also hit the woodwork as Leeds took charge early on.
Cresswell finally scored when he turned on Jemal Johnson's pass to fire home.
A late rally was inevitable and Luton almost levelled when Matt Heath felled Leon Barnett, only for Casper Ankergren to save Dean Morgan's woeful penalty.
Leeds boss Dennis Wise:
"We have to maintain that level of performance until the end of the season. I think that if we do that then we shouldn't be too far away.
"I think it would have been extremely harsh on us had they scored with the penalty at the end."
Luton boss Mike Newell:
"We didn't rattle them enough, we didn't defend well at times and we didn't keep the ball.
"But in the last 20 minutes we showed a lot of character and to miss the penalty was galling and I'm devastated.
"We have nine games left which is loads of time to get ourselves out of trouble. It is in our hands."
Leeds: Ankergren, Richardson, Heath, Michalik, Lewis, Carole (Armando Sa 88), Douglas, Kishishev, Blake (Moore 90), Johnson (Healy 85), Cresswell.
Subs Not Used: Stack, Kandol.
Booked: Johnson, Douglas, Michalik.
Goals: Cresswell 50.
Luton: Beresford, Keane, Carlisle, Heikkinen, Davis, Bell, Barnett, Robinson, Emanuel, Feeney (Morgan 69), Runstrom (Talbot 69).
Subs Not Used: Brill, Spring, Langley.
Booked: Emanuel, Runstrom.
Att: 27,138.
Ref: P Walton (Northamptonshire).