Germany's most successful club coach in terms of league title wins, Lattek was responsible for overseeing a golden period in Bayern's history, which saw them claim three successive Bundesliga crowns and the 1974 European Cup.
He went on to enjoy further success with Borussia Monchengladbach and Barcelona before returning to Bayern in 1983 and claiming a further five trophies.
Following Lattek's passing at the age of 80 in a nursing home in Cologne, Rummenigge spoke of the sorrow felt at Bayern.
He told the club's official website: "We are deeply moved and affected by the news of Udo Lattek's death. His name is so closely associated with the rise of FC Bayern in the successful 1970s.
"Udo Lattek was one of Germany's most successful football coaches. As well as this, he was one of the sport’s biggest personalities, at home and abroad, for many decades.
"We have lost one of FC Bayern's true greats, a personal supporter and friend."
Part of the coaching team that led West Germany to the final of the 1966 World Cup, Lattek initially took charge of Bayern in 1970, spending five years with the club.
He became the first man to lead a German team to European Cup glory in 1974, and went on to claim the UEFA Cup with Monchengladbach in 1979 and the European Cup Winners' Cup at Barca three years later.
Lattek, who won the German championship eight times in all, also coached Borussia Dortmund, Cologne and Schalke, his final assignment seeing him rescue the former from relegation in 2000.
Wolfgang Niersbach, president of the German Football Association, said: "Udo Lattek was already a living legend, he will be missed.
"He was not only the most successful coach in the Bundesliga history, but an outstanding personality of German football. He had a decisive influence over decades.
"In Udo Lattek football loses one of the most successful coaches and a great personality," added Monchengladbach managing director Stephan Schippers.
"His name is inextricably linked with Borussia success in the late 70s. Our sympathy goes to his family."
FIFA president Sepp Blatter paid his respects to Lattek via Twitter.
Blatter wrote: "I heard about Udo Lattek's death in deep sadness. A coaching legend is gone. My sympathy goes out to his family."
Barca president Josep Maria Bartomeu also tweeted: "Udo Lattek has left us. Barca is in mourning for the man who coached our team at the start of the 1980s. May he rest in peace."
