Kelly, who made his Leeds debut in 1991, hardly put a foot wrong, unlike some of his team-mates who struggled to cope with an uneven Elland Road surface.
After a disappointing first half, Leeds struck twice in five minutes with goals by Jonathan Douglas and Eddie Lewis. Luton claimed they should have had a penalty just before the opening goal when the ball struck Sean Gregan's hand, but referee Keith Hill waved play on.
And although striker Steve Howard pulled a goal back, climbing above Douglas to head into the net with seven minutes left, Leeds hung on and Luton had Sol Davis sent off just before the end for his second bookable offence.
Marlon Beresford rescued Luton by diving to his left to turn away an angled shot from Lewis and when David Healy robbed Chris Coyne just outside Luton's penalty box the Irishman crossed much too close to Beresford with Rob Hulse and Robbie Blake ready to pounce.
But the miss of the match arrived on 34 minutes when Kevin Foley contrived to head over the Leeds bar from inside the six-yard box. Croatian Ahmet Brkovic did much better, shooting inches past the foot of a post from 18 yards, but the half-time whistle came as a relief to everyone.
Leeds turned up the power after the break and after Blake had headed narrowly wide, Douglas finished off a neat passing move by cutting between two defenders and shooting past Beresford in the 49th minute.
Then Kelly's long ball forward was flicked on by Blake to Hulse, whose low cross from the left was turned in at point-blank range by Lewis.
Hatters manager Mike Newell responded by making a double substitution, replacing Dean Morgan and Brkovic with Enoch Showunmi and former Leeds apprentice Warren Feeney, while Leeds sent on Eirik Bakke for Healy.
Leeds withstood substantial late pressure and after Howard had cut the deficit with his 100th goal, as he forced a magnificent save from Neil Sullivan late on.