MeyGen is to install the tidal array in stages in the Pentland Firth, between Orkney and the Scottish mainland.
It will begin with a 9MW demonstration project of up to six turbines, with construction expected to take place on a phased basis until 2020.
When fully operational, the 86MW array could generate enough electricity to power the equivalent of 42,000 homes.
That is the equivalent of 40% of homes in the Highlands, the Scottish government said.
It will be the first commercial deployment of tidal turbines in Scottish waters.
Scottish-registered company MeyGen is a joint venture between investment bank Morgan Stanley, independent power generator International Power and tidal technology provider Atlantis Resources Corporation.
Its tidal energy project is located in the Inner Sound of the Pentland Firth off the north coast of Caithness.
The firm has agreed a 25-year lease with the Crown Estate for an area encompassing about 1.4 square miles (3.5 square kilometres) of fast flowing water between the island of Stroma and the north easterly tip of the Scottish mainland.
Its AR1000 turbine is claimed to be the world's most powerful single-rotor tidal device.
Each of the devices, which stand 22.5m (73ft) tall, weigh 1,500 tonnes and have a rotor diameter of 18m (59ft), could generate up to 1MWt of power.
Phase one of the plan would see 86 turbines deployed, with MeyGen hoping a second phase would eventually see up to 400 submerged turbines at the site.'Climate change'
Scottish Energy Minister Fergus Ewing said: "Today we have granted consent to MeyGen Limited to develop the largest tidal turbine array in Europe and the first commercial project off these shores.
"This is a major step forward for Scotland's marine renewable energy industry.
"This exciting development in the waters around Orkney is just the first phase for a site that could eventually yield up to 398MW."
Speaking before the Scottish Renewables Marine Conference got under way in Inverness, Mr Ewing also announced that developers Aquamarine Power Limited and Pelamis Wave Power are to share a slice of a £13m wave "first array" support programme.
The award is part of the Scottish government's Marine Renewables Commercialisation Fund.
Mr Ewing said the tide was turning for the wave sector.
He added: "We must tackle climate change. We need to reduce our reliance on fossil fuels through better and more efficient uses of energy.
"Marine energy - a home-grown technology with huge potential - is part of the solution."
The announcement was welcomed by environmental group WWF Scotland.
Director Lang Banks said: "This is a significant announcement and a major boost for the marine renewable industry in Scotland.
"However, as there is little point in generating huge amounts of marine renewable energy on Scotland's islands if it cannot also be got to the mainland, we now need UK and Scottish ministers to find a way forward that enables us to harness the full potential of this clean energy source.
"Alongside energy saving measures, marine renewables will have a critical role to play in helping Scotland reduce climate emissions as we phase out polluting fossil fuels and nuclear power.
"With careful planning we can harness Scotland's huge wave and tidal energy to help cut our climate emissions, while safeguarding the nation's tremendous marine environment."'Fraught with difficulty'
The Carbon Trust has estimated that wave and tidal resources could provide 20% of the UK's electricity if fully developed.
And the Scottish government believes the country's technological expertise in marine energy makes it extremely well placed to capitalise on domestic and overseas markets.
Scotland has been described as a Saudi Arabia of renewable energy potential, but developing power from offshore tidal streams is fraught with difficulty.
The harsh environment and extreme weather conditions make building, deploying and managing a fleet of tidal machines a treacherous challenge, as the BBC discovered when reporting on the emerging industry last year.