Trailing 1-0 from the first leg in Germany, and in the biggest game at Ibrox in 50 years, Gers skipper James Tavernier levelled the tie in the 18th minute with a close-range finish.
Amid a fever-pitch atmosphere, midfielder Glen Kamara drove in a terrific second from 20 yards just six minutes later and Rangers fans were in dreamland at the interval.
France striker Christopher Nkunku brought Leipzig back into it when he levelled the tie in the 70th minute with a smart volley but Gers midfielder John Lundstram fired in with 10 minutes remaining to book a place in the Seville final against Eintracht Frankfurt with a 3-2 aggregate win.
The frantic search for tickets, flights and hotels for Spain is now under way after Rangers reached their first European final in 14 years on a never-to-be-forgotten night in Govan.
It was Rangers most important game at Ibrox since they beat Bayern Munich 2-0 in the 1972 European Cup Winners’ Cup semi-final second leg on their way to winning the trophy, and the stadium shook beforehand with the noise of excited Gers fans.
After a minute’s silence before kick-off in tribute to popular kitman Jimmy Bell who died on Tuesday both sides battled frantically for control.
Visiting keeper and skipper Peter Gulacsi saved a long-distance drive from Jack in the seventh minute before Orban headed a corner wide at the other end.
However, right-back Tavernier, who scored twice against Braga in the previous round at Ibrox, struck the first blow, racing in at the back post to knock in a cross from Ryan Kent after good work by Kamara in keeping the move alive on the touchline.
More mayhem ensued when Aribo played in Wright to lay the ball off to Kamara who guided a left-footed shot from outside the box past the outstretched arms of Gulacsi.
Ibrox was in uproar.
Aribo then somehow mis-kicked from five yards out after being set up by Tavernier’s header from Borna Barisic’s deep cross.
Just before the break the makeshift centre-forward then fell to the ground after blocking a powerful free-kick from Angelino with his face and was replaced by Sakala.
With defenders Connor Goldson, Calvin Bassey and Barisic booked in the first half, the home side had to be cautious – yet retain a threat – as the Bundesliga outfit went in search of a tie leveller.
Wright and Jack made way for Arfield and Leon Balogun just before the hour-mark as the visitors began to dominate.
Gers keeper Allan McGregor made a great save from Konrad Laimer’s drive but moments later he had no chance when Angelino crossed from the left for Nkunku to volley in from 10 yards and the pendulum looked like it had swung towards the visitors.
But only until Lundstram pounced when Kent’s cross from the left was cleared to his feet 12 yards from goal and he gleefully drove it low into the net.
Rangers are on their way to Seville.
Additional reporting by PA Sport.