James Tavernier gave Rangers the lead with a 38th-minute penalty after a handball by Dan-Axel Zagadou from a Rangers corner.
Three minutes later, Dortmund again failed to cope with a Rangers corner. Joe Aribo nodded the ball to the far post where Alfredo Morelos was unmarked and scored from a few inches.
Borussia Dortmund named the same starting lineup that won comfortably at Union Berlin in the Bundesliga on Sunday but struggled to cope all evening.
At halftime, coach Marco Rose brought on Youssoufa Moukoko and Giovanni Reyna but it made no immediate difference.
Four minutes after the break, Ryan Kent, who tormented Dortmund all night, dribbled down the left and found John Lundstram who scored with a powerful strike for the third Rangers goal in the 11th minute
Two minutes later, Jude Bellingham replied for the home team with a rasping left-foot shot.
But two minutes after that, the Rangers scored again as Zagadou, attempting to deprive Morelos of another tap in, deflected the ball into his own goal.
Pedro Guerreiro replied with eight minutes left with a ferocious long-range left-footer.
Rangers survived without further scares to put themselves in a position to take revenge for a heartbreaking loss when the two teams last met, in the first knockout round of the competition, then known as the UEFA Cup in 1999.
They lost on penalties after the second leg in Dortmund. Current manager Giovanni van Bronckhorst was one of the Rangers players who missed from the spot.
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