The company has taken a 91% stake in the London-based We7, with the remaining shares to be purchased in the coming weeks. The overall deal price was £10.8m.
Peter Gabriel, the frontman of Genesis who founded We7 with Steve Purdham, the firm's chief executive, is thought to be in line to earn £2m from the deal.
Similarly to Spotify, We7 offers a library of 11m licenced music tracks for users to stream on multiple devices, but they can also create customised internet radio stations based on music they like, or might like based on their tastes.
The service is available online on PC and Mac, or via apps on iPhone, iPad and Google Android smartphone and tablets. The company makes its money from a mixture of advertising and user subscription models.
Tesco said that the acquisition is part of its strategy to "offer customers new and innovative ways of accessing digital entertainment".
This follows the firm's deal to acquire movies on-demand provider Blinkbox last year, which resulted in Tesco offering its 'Buy the Physical, Get the Digital' offer on DVDs and Blu-ray.
Tesco said that it plans to launch "additional digital music services from the WE7 platform in the coming months".
Mark George, the digital director at Tesco, said that both customers and technology are "transforming the way we listen to music".
"Tesco is already one of the UK's largest retailers of CDs; this move will help us offer a greater choice for the growing number of customers who want to access music instantly on any device, whenever and wherever they want," he said.
"WE7 has a great team and a good technology platform from which we can launch a range of digital music services in the future."
"Tesco has been an innovator in entertainment retailing for many years and we look forward to continuing this innovation digitally."
Alongside the announcement, Tesco said that it will increase its capital investment in online business to around £150m in 2012/13.
The We7 deal follows an announcement by rival Sainsbury's earlier in the week that it had bought HMV's stake in social books online retailer Anobii for just £1.
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