Connacht coach Michael Bradley said there is great excitement in the province about this game.
The Connacht forwards produced an electrifying finale but the 14 men of Leinster hung on.
Leinster now lead the league table by four points while Connacht are rotted to the bottom.
Connacht have been granted a meeting with Leinster, Munster, Ulster and IRFU officials next week.
Daniel Riordan's try helped edge Connacht 10-6 ahead but Kieron Dawson's quick reply restored Ulster's lead.
Connacht winger Eoin Griffin got a late consolation, adding to Ian Keatley's early penalty.
Former Ulster development player McComish had spells with Rotherham and Otley before joining Connacht in 2009.
The leaders did mount some late pressure but Connacht held on to secure a memorable triumph.
The nerveless Keatley slotted the kick and Connacht claimed a glorious and deserved win.
But Fionn Carr grabbed a Connacht try after the break although Freddie Pucciariello replied for Munster.
Connacht made a bright start and fly-half Keatley put the underdogs ahead with an early penalty.
Connacht's frustration grew, as did their penalty count, while Keatley missed a 67th-minute shot at goal.
Connacht got the ideal start when Lacey muscled over from close range in the seventh minute.
Sexton kept his nerve to coolly slot a 40-metre drop and dash Connacht's hope at the death.
"The Dragons were due to fly from Luton to Galway on Thursday, " added Connacht chief executive Kelly.
Leinster had to withstand some early Connacht pressure but managed to restrict the visitors to Keatley's penalty.
Connacht's frustration was compounded as flanker John Muldoon was yellow-carded for dangerous driving play at a ruck.
Cave went over as Connacht crumbled and Cunningham sprinted down the line before claiming the final try.
Connacht captain John Muldoon was sin-binned on his 100th cap for Hook's third penalty before the break.
Connacht fly-half Andy Dunne start at fly-half with Ulster's Niall O'Connor as his understudy on the bench.
John Muldoon has been released from the Ireland squad to play for Connacht against Samoa on Tuesday evening.
Paddy Wallace's two penalties gave Ulster a 6-3 lead at the interval with Tim Donnelly replying for Connacht.
The European champions led through Warwick's eighth-minute penalty but two Keatley kicks had Connacht ahead at the break.
Flanker Johnny O'Connor is another to return after some impressive displays for Connacht as is Ulster's Tommy Bowe.
Hogg then strolled over off the back of a scrum with an increasingly frayed Connacht down to 13 men.
Connacht has scored first through former Leinster man Andy Dunne who landed an 11th-minute penalty after two earlier failures.
Connacht had claimed an interval lead thanks to a try by Keith Matthews.
But it is expected that Connacht coach Michael Bradley will take temporary charge.
Connacht fly-half Tim Donnelly ended the dismal first half by striking a penalty to cut Ulster's lead to 6-3.
Connacht's problems began when full-back Troy Nathan was yellow carded for deliberately palming a Graeme Morrison chip into touch.
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