The iconic costume, believed to be the only one in a UK collection, will be one of a number of exhibits in the establishment’s Scottish Design Galleries.
Speedo was founded by a Scotsman, Alexander MacRae, who emigrated to Australia in the early 1900s.
But it was the classic Racerback costume, first launched in the 1920s, that put the brand on the international stage.
It caused moral outrage when it was first unveiled and was even banned on some beaches for exposing too much of the wearer’s flesh.
But its sleek, lightweight design, which allowed swimmers more freedom of movement and reduced drag, meant the suit quickly became a favourite with athletes.
The new design was also marketed to surfers and sunbathers, with a Speedo catalogue describing the Racerback as giving “maximum body exposure” for those in search of a tan.
The pioneering suit turned the company into a household name and paved the way for future innovations.
Meredith More, assistant curator at V&A Dundee, said: “We are delighted to be able to include such an early Speedo swimsuit in our Scottish Design Galleries.
“The Racerback’s revealing back straps challenged moral codes in the 1920s, when mixed bathing was only just becoming acceptable, but nobody could deny his hydrodynamic design allowed swimmers to achieve faster times.”
Swedish swimmer Arne Borg, who won five Olympic medals and broke 32 world records, was one of those who embraced the daring new design and featured in several Speedo advertisements.
The firm also designed swimsuits for women. In 1932 Claire Dennis was almost disqualified from the Olympics in Los Angeles for wearing a costume deemed to show too much shoulder.
MacRae, born in 1888, grew up in a small fishing village near Loch Kishorn in the West Highlands.
He moved to Sydney in 1910 and set up a hosiery company called MacRae Knitting Mills four years later.
The firm was supplied the Australian army with socks during the First World War before branching out to cater for the growing popularity of beach sports.
At the time swimsuits were made of wool and had sleeves to protect the wearer’s modesty. Instead, the Racerback had straps that crossed at the back and was made of cotton or silk, which absorbed less water.
Both the radical design and these materials enabled freer movement for the swimmer and so greater speed.
The design was also significantly more tight-fitting than other swimwear of the time and included the distinctive Speedo tick logo.
Under MacRae’s leadership, Speedo caused further controversy when it dressed the Australian men’s team in swimming shorts instead of the traditional one-piece suit for the 1936 Olympic Games in Berlin.
The 100-year-old swimsuit, which has never been worn, is on loan to V&A Dundee from Leicestershire County Council Museums Service.
It was bought for the collection after being discovered in a charity shop, with the manufacturer’s labels still attached.
The V&A Dundee opens on 15 September.