Aleksandar Mitrovic opened the scoring for the Cottagers in the first half, tapping in from close range after a header from Tosin Adarabioyo was cleared off the line.
In the second half, Town stepped up a gear and took the game to Fulham. Their equaliser came just after the hour mark, with Adebayo heading in from James Bree’s free-kick.
Nathan Jones made two changes to the Hatters side that were 3-0 winners away at Blackpool last weekend.
After impressing off the bench at Bloomfield Road, Admiral Muskwe came into the starting line-up for his fourth league start of the season, with Carlos Mendes Gomes dropping out. Pelly-Ruddock Mpanzu also returned, starting ahead of Henri Lansbury.
James Shea kept his place in goal for Town, with Simon Sluga named on the bench after missing last weekend’s game due to illness.
The game’s first effort on goal came from the visitors after seven minutes. Tom Cairney found Joe Bryan, whose effort flashed wide of Shea’s post.
Fulham would go ahead just before the 20-minute mark. Harry Wilson’s free-kick was met by the head of Tosin Adarabioyo, who saw his effort cleared off the line by Sonny Bradley, but Mitrovic reacted quickest to tap in from close range.
As the first half went on, Town seemed to grow into the game but just lacked the final pass. Bree’s ball into the box was good, but no Luton player could get on the end of it.
Muskwe then came close for the Hatters. Breaking into the Fulham area and cutting inside of Tim Ream, his venomous effort went just wide of Marek Rodak’s near-post.
Jones’ side started off the second half on the front foot, with Muskwe involved again. Twisting and turning inside the 18-yard box, his effort didn’t trouble the Fulham goalkeeper, firing wide.
The Hatters threatened the Fulham goal once more early on in the second half. Both Elijah Adebayo and Jordan Clark saw their efforts blocked, before Reece Burke’s cross was headed clear at the back-post.
And Town would have a deserved equaliser just after the hour mark. James Bree delivered a brilliant free-kick into the Fulham area, and Adebayo headed beyond Rodak for his 10th goal of the season.
Fulham came close to regaining the lead in fortuitous circumstances. Town didn’t clear their lines from a corner, as Muskwe’s wayward header forced Shea into a smart stop.
Jones made a double switch with just under 15 minutes remaining. Fred Onyedinma and Gabe Osho were introduced for Muskwe and Allan Campbell.
Bree had a chance to put the Hatters ahead in the last few minutes of the game. Kal Naismith went on a surging run, beating a number of Fulham defenders, before finding Bree, who couldn’t find the far corner with his effort.
In front of a sold attendance of 9,992, but with fewer than 9,000 actually inside the ground, the old place rocked with the league leaders on the ropes at times during the second half.
It would end with a score draw at the Kenny, after Luton’s second-half performance put Fulham to the sword.
Goals:
Town: Adebayo 62
Fulham: Mitrovic 19
Att: 9,992 (1,029 away)