The deal between the two firms should have been completed earlier this year but it was delayed by a hold-up in regulatory approvals.
The sale will see the end of production of mobile phones by Nokia.
“Today we welcome the Nokia devices and services business to our family,” said Microsoft chief executive Satya Nadella.
“The mobile capabilities and assets they bring will advance our transformation.”
The Finnish company will now focus on networks, mapping services and technology development and licences.
Two Nokia plants will remain outside the deal – a manufacturing unit in Chennai, India, subject to an asset freeze by Indian tax authorities, and the Masan plant in South Korea, which it plans to shut down.
Former Nokia chief executive Stephen Elop has become executive vice president of the Microsoft devices group, in charge of Lumia smartphones and tablets, Nokia mobile phones, Xbox hardware, Microsoft Surface, and Perceptive Pixel (PPI) products.