John Lundstram's strike sparked bedlam at Ibrox as Rangers progressed 3-2 on aggregate
Trailing 1-0 from the first leg, Rangers turned the semi-final on its head within 25 minutes through goals from James Tavernier and Glen Kamara.
Christopher Nkunku's 32nd goal of the season levelled the tie again before John Lundstram shook Ibrox with 10 minutes left after sweeping home.
Leipzig, despite their talented squad, could find no answer in the closing stages amid a cauldron of noise as a depleted Rangers side completed their remarkable run from the play-off round to the final.
Giovanni van Bronckhorst's men will meet Eintracht Frankfurt in Seville on 18 May with a tantalising chance to win the club's first European trophy for 50 years.
When the full-time whistle arrived, the Rangers players looked on in utter disbelief at their incredible achievement, but it was a performance worthy of winning any European tie.
Using the emotion of Ibrox, as they have throughout the campaign, just days after the passing of club icon and kitman of 30 years Jimmy Bell, they stormed out the blocks and into a 2-0 lead despite the absence of key strikers Alfredo Morelos and Kemar Roofe.
Firstly, Kamara held up a long ball he had no right to retain, and offloaded to Ryan Kent who picked out Tavernier at the back post for his seventh goal of the Europa League campaign.
The fans behind the goal cascaded towards the pitch in sheer elation, but it got even better six minutes later when Kamara caressed a shot with his weaker foot into the back of the net.
Leipzig, a club with a turnover about six times that of Rangers, were completely rattled and should have fallen three goals down before the break as Joe Aribo sclaffed a volley in the air from five yards.
That came so close to capping a perfectly executed first-half display from Rangers, and after the break they showed the guts and determination that has characterised their run to the final.
After seeing Aribo and Ryan Jack limp off injured, they were picked apart by Angelino's delicious cross from the left and Nkunku's gorgeous first-time finish and the momentum seemed to have swung away from Van Bronckhorst's side.
But despite the Germans cranking up the pressure and legs tiring, Rangers somehow got themselves up the pitch.
The lively Kent ghosted past his man and hung a cross into the box which almost dropped in, and Lundstram was lurking to drill home the clearance for the winning goal to send Ibrox rocking from its foundations.
Fashion Sakala should have put it to bed in stoppage time when clean through, but it did not matter as Rangers saw out the final minutes in relative comfort having knocked the stuffing from Leipzig, who managed only two shots on target all night.
-- Courtesy of BBC Sport