Legendary producer Quincy Jones introduced the odd American couple, praising Gaga as a singer who had "blown his mind." He also recalled working on the musical arrangement for Bennett's 1966 hit "Girl Talk."
"This would be Claude's definite dream," Jones said, referring to Claude Nobs, the late founder of one of Europe's most prestigious summer music festivals, now in its 49th edition.
Bennett, the only person in the famed Stravinski Auditorium wearing a jacket and tie, and the blond Gaga, appearing first in a long shiny silver dress with plunging neckline, opened with "Anything Goes," a Cole Porter tune of the 1930s.
The pair of New Yorkers then sang "Cheek to Cheek" by Irving Berlin, the title track from their 2014 album of jazz standards for which they won a Grammy Award in February.
Bennett, just weeks shy of his 89th birthday, and Gaga, already a six-time Grammy winner at age 29, delivered 30 songs during their 100-minute non-stop show, backed by nine musicians.
"We're selling our album around the world, please buy it. She really needs the money," Bennett quipped.
Gaga, born Stefani Joanne Angelina Germanotta, is better known for her hits "Paparazzi" and "Poker Face."
Gaga hit the high notes in Montreux in half a dozen eye-popping outfits, changing backstage during Bennett's solos. During love ballads, she gave him hugs or kisses on the cheeks.
For their "I Won't Dance" duet, she wore a black dress and long gloves, with glittering eyebrows.
In "I Can't Give You Anything But Love," Gaga played the vamp in a red feathered dress with transparent top. For "Let's Face the Music and Dance" her silver dress with chain belt gave her the allure of the swinging 60s.
She sat back-to-back with pianist Michael Renzi cabaret style for "Everytime We Say Goodbye," another Cole Porter hit.
Sinatra centenary
Bennett paid tribute to Sinatra, born 100 years ago, saying the late star had been "opening his soul" with the song "I've got the World on a String." He then performed his personal favourite, "In the Wee Small Hours."
Introducing the song "Smile," Bennett told the crowd that he had once received a fan letter from Switzerland about the best-selling theme song from the 1936 movie "Modern Times."
"I couldn't believe it. It was from Mr. Charlie Chaplin," he said of the British actor who lived in nearby Vevey.
Gaga, dressed in a vintage pink satin dress with huge bow at the waist, sang "La Vie en Rose," earning the biggest applause of the night for her moving version of Piaf's signature song.
