Trailing 1-0 from the first leg in Germany, and in the biggest game at Ibrox in 50 years, skipper James Tavernier levelled the tie on aggregate in the 18th minute with a close-range finish.
Midfielder Glen Kamara drove in a terrific second from 20 yards just six minutes later and the Gers fans were in dreamland at the interval.
The in-form 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.
Amid unbelievable scenes at Ibrox at full-time, Rangers legend Ally McCoist was seen joining in the celebrations…
He said on BT Sport at full time: "I can't tell you how proud I am of that team.
"I've been fortunate enough to come to this place and I've seen victories over Bayern Munich, I've seen victories over Dynamo Kiev, I've seen victories over Leeds United but tonight's atmosphere has been the best - the best!
"That tonight will take some beating.
"Lundstram, [Calvin] Bassey…dear me. To a man, every one of them should be so proud of themselves.
"Nine years ago, we were playing East Stirling in the Third Division, now we're in a European final!"
And former Scotland international Charlie Adam suggested there will be around 100,000 Scots heading to Spain on May 18.
"The supporters have been fantastic," Adam said on talkSPORT.
"There's going to be a fair few of them in Seville - I reckon 100,000!"
It has been an epic journey for Rangers who were playing their 18th game in UEFA competitions this season against Leipzig.
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.
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.
Aribo then somehow mis-kicked from five yards out after being set up by Tavernier's header from Borna Barisic's deep cross.
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.
But then 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.
They will face Frankfurt, who beat West Ham in the other semi-final, in Seville.