"My record of 85 goals in 60 matches, which stood since 1972, was beaten by the best player in the world, Lionel Messi, and I'm very happy for him," Müller said, according to a spokesman for his former club, Bayern Munich.
The former Germany striker added that he thought Messi was "an incredible player, a giant who is at the same time very nice and a polished professional".
Messi, still just 25, on Sunday equalled then surpassed Müller's record with two first-half goals that secured a 2-1 win for his side Barcelona against Seville club Real Betis, earning him plaudits around the world.
"I hope he'll score one or two goals again in the matches that remain in 2012 and that his record will stand for the next 40 years," added Müller, who was nicknamed "The Bomber."
Müller, who was at his peak between 1964 and 1979 but battled alcoholism since his retirement in 1981, joked that Messi only had one fault: "He doesn't play for Bayern Munich."
Messi has previously beaten the record for the most goals in Spain's La Liga in a season -- 50 -- and is favourite to win an unprecedented fourth Ballon d'Or award in succession when the prize is announced on January 7.
At 67, Müller is by far the best goalscorer in the history of the Bundesliga with 365 in 427 matches but Miroslav Klose is one short of equalling his international goals record for the Mannschaft of 68.