The Cottagers snatched a share of the spoils with their third equaliser in the fourth minute of stoppage time.
That late, late strike, coupled with Stoke City's last-gasp winner in their clash with Sheffield Wednesday, means Graeme Jones' side dropped into the bottom three of the Sky Bet Championship.
To see how the Hatters players rated on the day, see below.
9. James Collins 7.5
Led by example from the front, never giving Fulhams defence a moments peace. Took his goal well and was unlucky not to get a second due to the linesmans flag | Getty
10. Luke Bolton 6
Flapped at a couple of high balls, which let Fulham attack, most notably when Bobby Decordova-Reid ran in on goal, where Simon Sluga turned sweeper to get him out of jail | Getty
11. Kazenga LuaLua 8.5 (Star man)
LuaLua scored Lutons openerand then preceded to give Fulhams full backs a torrid time, so much so that had to kick him up in the air to stop him | Getty
12. Harry Cornick 7
Made in impact and took his goal well, but should have made it two late in the game, to make it 4-2, having done the hard work of beating Tim Ream, but pulled his shot wide | Getty