Luton led three times with goals from Kazenga LuaLua, top scorer James Collins and substitute Harry Cornick, but each time the Cottagers hit back, levelling through Bobby Decordova-Reid and Aleksandar Mitrovic before the former broke Town hearts with a third equaliser four minutes into injury-time.
Jones made three changes to his starting line-up, with Ryan Tunnicliffe, Callum McManaman and LuaLua all returning to the starting XI in place of Jacob Butterfield, Andrew Shinnie and Harry Cornick.
The Town made a dream start on four minutes when Collins LuaLua pressed Fulham as they tried to play out in their own six-yard box. LuaLua won the ball from Ream bang in the centre of the goal, it rolled to Collins who controlled it back into LuaLua's path to stab home past Rodak, who got a touch but couldn't keep it out.
The visitors were level within five minutes when Ivan Cavaleiro crossed from the right and Decordova-Reid was allowed the freedom of the Hatters' penalty area to head past Simon Sluga.
Collins soon sent a glancing header just wide as Town continued in a positive vein, but they had to call on Sluga to race off his line and clear with his feet when Bolton's short backpass put Cavaleiro in the clear.
The Croatian keeper was soon turning a Joe Bryan 20-yarder over the bar before making a terrific one-handed save low to his right to keep out a Tom Cairney drive.
LuaLua was enjoying himself up against Cyrus Christie and in the 2st minute he sent one of several low crosses dangerously across the six-yard box that went begging.
Cairney tried his luck again in the 25th minute, but Sluga gathered on the skiddy surface at the second attempt.
The Town's lead was restored just before the half-hour when Berry's free-kick from deep on the right found James Bree challenging Alfie Mawson, and the ball spurted out to Collins, who controlled and finished into the top corner from eight yards out.
Collins thought he had his ninth of the season on 36 minutes when he met another Berry free-kick, this time from the left, and turned it in off a post, but a linesman's flag ruled this one out.
It was the best 45 minutes of the season from the Hatters at Kenilworth Road, and Jones side deservedly headed in at the break with a 2-1 lead.
We were into the 90th minute when Cornick pounced on another mistake by Ream and raced clear, but his shot across Rodak also beat the far post.
Fulham would deal that late blow when Cordova-Reid followed up and turned in the rebound from another Sluga save to salvage a point for the visitors.
Coupled with Stoke's win at home to Sheffield Wednesday, it meant the Town dropped into the relegation zone on goal difference.
But as the final home game of a decade that kick-off here at Kenilworth Road with a 3-2 defeat to Ebbsfleet United in the Conference, the reality is that we go into that cherished year of 2020 with so much to be hopeful - and thankful - for.
TOWN: Sluga, Bolton, Pearson, Bradley (c), Bree, Tunnicliffe, Mpanzu, Berry, McManaman (Cornick 65), LuaLua (Jones 76), Collins.
Subs: Shinnie, Moncur, Butterfield, Sheehan, Stech.
Goals: LuaLua 4, Collins 28, Cornick 84
FULHAM: Rodak, Mawson, Mitrovic, Cairney (c), Ream, Decordova-Reid, Cavaleiro (Kebano 76), Reed (Knockaert 69), Christie, Bryan (Kamara 89), Onomah.
Subs: Bettinelli (GK), Johansen, O'Riley, Sessegnon.
Goals: Decordova-Reid 9, 90+5, Mitrovic 77
REFEREE: Dean Whitestone
