Captain James Tavernier and Glen Kamara struck in the first half to give the Scottish champions a two goal cushion and quickly cancelled out the Germans narrow first-leg lead.
Christopher Nkunku levelled the tie on aggregate against the run of place but the irresitable John Lundstram slammed home in the closing stages to send his side to Spain on May 18.
Fellow Bundesliga outfit Eintracht Frankfurt now lie in wait as Rangers aim to seal European silverware.
GlasgowWorld rates how Giovanni van Bronckhorst’s heroes performed:
ALLAN McGREGOR - 7
Enjoyed a largely uneventuful night due to the protection of his defensive line in front of him. Produced an excellent save when called upon to deny Laimer but was beaten moments later.
JAMES TAVERNIER - 9
A talismanic figure in this Rangers side. Led by example once more and broke the deadlock after 18 minutes to ensure his status at top scorer in the competition this season.
CONNOR GOLDSON - 7
Looked to be in for a tough night up against Nukunku and Yussuf Poulsen and picked up an early caution. Lost Nukunku for Leipzig’s goal but stood firm in closing stages.
CALVIN BASSEY - 8
Another epic display from the imposing Nigerigan. Won numerous aerial battles and was booked in the first-half for pulling back a Leipzig player. Solid as a rock.
BORNA BARISIC - 6
Worked hard down the left-hand side but conceded a silly free-kick on the stroke of half-time with a needless challenge.
JOHN LUNDSTRAM - 10
Deployed in a back three during the first-half. Another sublime performance before moving into midfield after the break and converted the biggest goal of his career.
RYAN JACK - 7
Played a vital role in the midfield battle with a disciplined performance on his 150th appearance for the club. Limped off injured just before the hour mark.
GLEN KAMARA - 8
Arguably one of his best performances of the season. The Finland international started the move that led to Rangers’ opening goal by displaying great strength. Steered a brilliant left-foot shot into the bottom corner to stretch the lead.
SCOTT WRIGHT - 8
Much improved display after toiling for large spells as Rangers’makeshift striker in Germany last week. Helped limit Angelino’s influence on the game and seemed to thrive on the occasion.
JOE ARIBO - 6
Held the ball up well but missed a great chance to make it 3-0 when scuffing his effort over the bar from just a few yards out. His game ended abruptly after sustaining a head knock.
RYAN KENT - 9
Had a leading role to play once again and his delivery for Tavernier’s opener was first class. Deep cross led to Lundstram’s matchwinner in the 80th minute.
FASHION SAKALA - 6
His pace caused the tiring Leipzig defence problems. Worked hard to help run the clock down.
SCOTT ARFIELD - 5
Replaced Scott Wright and almost got an assist after displaying good footwork near the byline.
LEON BALOGUN - 5
Brought on for Ryan Jack in a defensive reshuffle as Van Bronckhorst reverted to a back four. Helped stem the tide late on.
Unused: Jon McLaughlin (GK), Robby McCrorie (GK), Leon King, Steven Davis, Aaron Ramsey, James Sands, Alex Lowry, Amad Diallo