The 13.6-kilometre (8.5 miles) tunnel includes an immersed tube tunnel which government officials say is the world's deepest at 60 metres (nearly 200 feet) below the seabed.
The inauguration of the ambitious project that has cost an estimated three billion euros coincides with the 90th anniversary of the founding of modern Turkey.
"Turkey will celebrate two feasts together," Transport Minister Binali Yildirim said earlier this month.
The tunnel is part of a larger "Marmaray" project that also includes an upgrade of existing suburban train lines to create a 76-kilometre (47-mile) line that links the two continents.
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The idea was first floated by Ottoman sultan Abdoul Medjid in 1860 but technical equipment at the time was not good enough to take the project further.
However the desire to build an undersea tunnel grew stronger in the 1980s and studies also showed that such a tunnel would be feasible and cost-effective.
His ambitions were one cause for the massive anti-government protests that swept the country in June, with local residents complaining that the premier's urban development plans were forcing people from their homes and destroying green spaces.
Japan's Prime Minister Shinzo Abe will be present at the official opening, as the Japan Bank for International Cooperation was the main financer contributing 735 million euros (USD 1 billion) to the project.
Some 40,000 objects were excavated from the site, notably a cemetery of some 30 Byzantine ships, which is the largest known medieval fleet.
But these unexpected finds eventually frustrated Erdogan, who complained two years ago that artefacts were trumping his plans to transform Istanbul's cityscape.