According to his family, the 85-year-old died on Monday following a stroke.
White was chosen ahead of Ringo Starr in September 1962 to play drums on the single version of Love Me Do and its B-side, P.S. I Love You.
White, who was born in Glasgow in 1930, is also believed to have played on the album version of Please Please Me.
He could therefore legitimately claim to be one of the so-called "Fifth Beatles", alongside the likes of Pete Best, Stuart Sutcliffe and others.
White also played drums on Lulu's 1964 cover of Shout and Sir Tom Jones' 1965 single It's Not Unusual.
He went on to tour with Marlene Dietrich, Burt Bacharach and Rod Stewart and perform with the BBC Scottish Radio Orchestra.
In a 2009 interview with a New Jersey newspaper, White revealed he was often called to London's Abbey Road studios in the 1960s.
"I would get a call from EMI and you never knew what you were going to be asked to do," he told The Progress.
White was paid a one-off fee - Â£5 - for his three hours with the Beatles and received no subsequent royalties.
Starr, who played drums on earlier versions of Love Me Do, can be heard playing the tambourine on the White recording.
"You could tell it was something different and very special," White told the Daily Record in 2012. "But I didn't know just how special it would become."
In later life White emigrated to the US, where he married the voiceover actress Thea Ruth and became a drumming instructor.
The New York Metro Pipe Band, one of the bands with whom he worked, described him on Facebook as an "all round gentleman".
He was also called upon to teach drums to musician turned actor Steven van Zandt for his role as Silvio Dante in TV hit The Sopranos.
Speaking to the BBC on Wednesday, White's family paid tribute to his "amazing humility and humbleness about his many achievements".
Andy White, the Scottish studio session musician who played the drums on Love Me Do and other early tracks by The Beatles, has died in New Jersey.