The two-time European Footballer of the Year winner feels that it would only be right if die Roten lifted the trophy after losing two finals in the past three years.
The Bavarians have not lifted the trophy since 2001, having lost two finals in the past three seasons, against Inter in 2010 and Chelsea 12 months ago.
Consequently, Rummenigge feels that Bayern has paid its dues and that it would only be right if they beat Borussia Dortmund to the title at Wembley a week from Saturday.
"We want to avoid a third final defeat at all costs," the Roten CEO told Sport Bild.
"This is for all of us and the soul of this side...[Bayern] deserve to receive this trophy. The team have worked so hard for it over the last five years."
Having already wrapped up the Bundesliga title with a record-breaking six rounds to go and, with a DFB-Pokal final still to come, Bayern is perfectly primed to record a treble.
Rummenigge believes that if Jupp Heynckes' men pull it off, they deserve to be ranked alongside the side that won three successive European Cups four decades ago.
"This team would go down in history," the former West Germany international enthused. "They would no longer be in the shadow of the team of the 1970s."
Rummenigge also denied that treble success would make next season impossible for incoming coach Pep Guardiola.
"The clocks go back to zero on July 1 [when the former Barcelona coach officially takes over from Heynckes]," he insisted. "But we can just try to win even more next season!"
It has been reported that Guardiola has already deemed Bayern winger Arjen Robben surplus to requirements, but Rummenigge insists that there is no chance of the Dutchman being offloaded this summer.
"Arjen has a contract with us so the prospect of a farewell at the end of the season is not an issue," he declared.
Rummenigge is also predicting a new contract for France international Franck Ribery, who recently revealed his desire to finish his career at the Allianz Arena.
"We have taken note of his words," he explained. "This is a sign of how well he has done and how he feels here.
"He is a great player and I can easily imagine him extending his contract."
Bayern concludes its Bundesliga campaign away to Borussia Monchengladbach this weekend.