Goals from James Tavernier, Glen Kamara and Lundstram ensured Rangers moved to the final for the first time since 2008, where they will face Eintracht Frankfurt, bidding to win a second European trophy 50 years after the 1972 Winners' Cup.
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Captain Tavernier opened the scoring in the 18th minute, becoming the Europa League's top scorer this season with his seventh goal, before midfielder Kamara's long-range effort six minutes later extended the lead.
Leipzig, who won the first leg 1-0, hit back when forward Christopher Nkunku received a perfect cross from Angelino to score in the 70th minute but Rangers claimed the vital goal when a defensive mix-up after a corner allowed Lundstram to sweep the ball home.
"It's unbelievable. Towards the end Lunny [John Lundstram] came up with the goods. A European final, it's what you dream of," Tavernier told BT Sport.
"We'll go there [final] full of confidence. Frankfurt got there for a reason, but it's one game and we'll fully back ourselves. We're in this to win it. We want to make all the fans proud."
Frankfurt are 11th in the Bundesliga and Rangers have already knocked out the German league's second-placed Borussia Dortmund and fifth-placed Leipzig this season.