Brian McDermott’s side - who finished the season with six successive league losses - thought they had battled their way to a hard-fought elimination play-off victory when Jamie Jones-Buchanan dived over for Kevin Sinfield’s kick to put them 20-18 in front with 12 minutes to go.
But there was late drama as seventh-placed Catalan - who had never before won here in nine previous visits - scored in the most surreal circumstances at the death.
Zak Hardaker had just cleaned up a threatening French kick and scrambled his way out from behind his own line to avoid conceding a drop-out.
There could have been a claim the England full-back actually performed a voluntary tackle to do so. Nevertheless, when he played the ball just in the field of play, captain Sinfield passed to Briscoe but his winger, inexplicably, dropped possession before any defender got near him.
Sam Williams, the Australian scrum-half who had been so influential for Catalan, picked up the gift and dived over to send their elated players and smattering of supporters into utter ecstasy.
The French side, notoriously poor travellers who had won just four of their 15 away games this term, fully deserved their victory given the manner in which they refused to yield against a Leeds side so accustomed to reaching Old Trafford.
They will no visit Huddersfield Giants on Friday for a place in the semi-finals while their beaten opponents will look back on a season where they finally ended their long wait for the Challenge Cup but endured their worst league finish since 1996.
After Warrington almost came unstuck against play-offs newcomers Widnes Vikings earlier today - they trailed 18-0 at home before eventually getting through 22-19 and setting up a Thursday eliminator at Castleford - Leeds had been warned about the potential of a shock.
They had led 14-6 at the break but had been thankful for an embarrassing error from Morgan Escare just before the interval for that lead.
The dazzling French full-back has earned many plaudits this season and, indeed, it was a surprise to some that he did not earn the Dream Team nomination ahead of Leeds’ own Hardaker.
But - after a Jamie Peacock fumble had given Catalan one last attacking opportunity in a tight opening period - he crabbed across the field 40m from his own line, throwing a dummy but only fooling himself as he inexplicably dropped possession with no one near.
Briscoe got a foot to it and then Kallum Watkins twice hacked on to beat a lethargic Vincent Duport to the touch down.
Sinfield improved and Leeds had their noses in front.
The Leeds captain had twice needed treatment after late challenges from Catalan defenders who clearly seemed intent on ruffling the ex-England captain.
After Louis Anderson caught him late following a pass in the 16th minute, Carl Ablett surged through from the resultant penalty to put the hosts in front.
Sinfield converted but Catalan grew into the game and made a couple of threatening breaks to show their potential.
Hardaker was called upon to drag down Zeb Taia and, soon after, it needed a timely tackle from Tom Briscoe to deny the Australian second-row .
Raiding Catalan scrum-half Sam Williams also found space and, so, it was no surprise when the visitors did level following an intricate passing movement that cut Leeds open again in the 25th minute.
It was Williams who finished after Benjamin Garcia had found Gregory Mounis on a perfect angle and Thomas Bosc improved.
However, when Duport was late flinging himself at Sinfield when the Leeds stand-off kicked, Hardaker took on kicking duties to put the home side 8-6 ahead five minutes before the break.
The second half was similarly fraught but Catalan came back into when Zeb Taia got in front of Briscoe to win the challenge for Bosc’s kick, who converted the 46th minute effort.
They took the lead for the first time 10 minutes later when Louis Anderson charged on to Williams’ lovely pass and barrelled through Hardaker’s desperate attempted tackle for Bosc to convert once more.
The inevitable pressure came from Leeds, understandably given they have been champions four times in the last six years.
It was no surprise then, that after Duport cheaply fumbled playing the ball, Jones-Buchanan duly scored after some terrific handling from Hardaker, Danny McGuire and Sinfield.
But there was some late heroics to come and, instead of the last time they met here in June when Leeds snatched a win after the hooter, it was Catalan benefiting from all the drama.
Leeds Rhinos: Hardaker, T Briscoe, Watkins, Moon, Hall, McGuire, Sinfield, Leuluai, Burrow, Peacock, Jones-Buchanan, Ablett, Delaney. Subs (all used) Aiton, Bailey, Achurch, Sutcliffe.
Catalan Dragons: Escare, Oldfield, Pomeroy, Duport, Millard, Bosc, Williams, Elima, Henderson, Anderson, Taia, Whitehead, Mounis. Subs (all used) Lima, Bousquet, Pelissier, Garcia.
Referee: Phil Bentham (Warrington).