James Tavernier and Glen Kamara turned the tie around inside the first 24 minutes at an emotionally charged Ibrox as players and fans united to pay tribute to long-serving kitman Jimmy Bell, who died on Tuesday.
Nkunku hit back for Leipzig in the second half, but John Lundstram scored the winner nine minutes from time.
Giovanni van Bronckhorst's men will face Eintracht Frankfurt in the final in Seville on May 18 in just their second European final in 50 years. The Scottish champions lost to Zenit St Petersburg in the UEFA Cup final of 2008, but have had to start again from the fourth tier following a financial collapse in 2012.
Last season they ended Celtic's run of nine league titles in a row and will return to the Champions League for the first time since 2010 with victory over Frankfurt at the Ramon Sanchez Pizjuan in 13 days' time.