The Scottish champions had lost their semi-final first leg to Leipzig 1-0 but hit back on home turf, with first-half goals from captain James Tavernier and Glen Kamara.
Leipzig squared the tie with Christopher Nkunku’s 70th-minute goal but the Gers got their winner from John Lundstram in the 80th minute.
Rangers, who were UEFA Cup runners-up in 2008, came through the Europa League play-off round to reach the May 18 decider, knocking off Borussia Dortmund, Red Star Belgrade, Braga and Leipzig along the way.
Their triumph comes only days after the death of club icon and kitman of 30 years Jimmy Bell.
“It’s very hard to find the words,” Gers boss van Bronckhorst told BT Sport. “It’s been an amazing night.”
“We said before the game we’d do everything possible. The players were fantastic. You can’t write a script better than this. We’re all very proud. We toasted inside to Jimmy for this one.
“The two opening goals were fantastic, just like we wanted. We knew Leipzig would come at us, then to score in the last 10 minutes gives you an enormous boost.”
Elsewhere, Eintracht secured their spot in the final with a 1-0 home win over West Ham United after Rafael Santos Borre’s 26th minute.
Aaron Cresswell was sent off in the 19th minute for the Hammers, while manager David Moyes was also sent to the stands after lashing out at a ballboy, by kicking a ball at him on the sidelines.
“I kicked a ball back at the ballboy so I apologise for that,” Moyes said. “He threw it very softly at me.”
He added: “If were being honest, we probably lost the tie in the first minute at the game at London Stadium, when we conceded a goal after 50 seconds which was rubbish, and since then weve been chasing the tie really.”