Both sides knew a win would see them into a clash with London Spirit on Friday for the right to face Trent Rockets in the final at Lord's.
Little returned the best figures in the men's Hundred, taking 5-13 from his 20 balls, while Sam Billings' half-century guided the visitors to a competitive 9-143.
Madsen's unbeaten fifty and late heroics from Paul Walter saw Originals home with a ball to spare.
The home side were brilliant in the field and the tone was set by Walter, who took a stunning diving catch in front of Old Trafford's party stand to dismiss Rilee Rossouw for 17.
Little backed that up with a superb set, taking the wickets of Jordan Cox and Will Jacks to leave the Invincibles 3-23.
Billings and Sam Curran set about rebuilding patiently, taking the score to 64 at the halfway mark.
Curran then launched a huge six off Tristan Stubbs but fell for 24 to end the blossoming partnership at 56.
Billings pushed on undeterred, smashing two sixes off Matt Parkinson's second set and passing his fifty from 30 balls.
With 20 balls remaining, Invincibles were 4-115 but the Originals finished strongly.
Billings fell to a brilliant catch from Tom Hartley to give Little his third before two smart bits of work from wicketkeeper Phil Salt - running out Tom Curran and correctly reviewing an edge off Hilton Cartwright - kept the Originals on top.
Little completed his superb five-wicket haul when Matt Milnes was caught for a duck and Gleeson struck in the final set to restrict the Invincibles to 143.
Manchester flew out of the traps, racing to 34 from just 16 balls before Laurie Evans fell for nine - hitting Aussie legspinner Peter Hatzoglou straight to midwicket.
They were 0-43 at the end of the powerplay but three balls later Salt attempted to hit Danny Briggs over the ropes and only found Cox.
Madsen and Stubbs set about rebuilding - with a Stubbs six helping the hosts to 2-74 at halfway, needing another 70.
Stubbs picked out Sam Curran to hand Briggs a second, but Madsen brought up a 34-ball half-century to reduce the target to 33 from 20 balls.
Tom Curran's tight set had the Originals' nerves building, before Perth Scorchers' Ashton Turner hit Sam Curran for 14 from his five balls to seize control again.
But Tom Curran went for just two from the penultimate set, with Turner also run out, to leave the Originals needing 11 from five balls.
Walter launched a huge blow which looked set to be caught, but Jacks could only tip the ball over the rope for six.
It proved the key moment as Walter struck the winning run two balls later.
