Ukraine has condemned the project, which cost 228 billion rubles ($3.7 billion).
European Union and U.S. sanctions have targeted those involved in the realization of the bridge, particularly businessman Arkady Rotenberg, a close ally of Putin whose company Stroygazmontazh won the construction contract.
Because of blocks imposed by Kiev, in addition to Western sanctions, a large amount of food is now shipped to the peninsula from Russian Federation, so the bridge will play an important role in reducing the region's reliance on sea transport.
Russian President Vladimir Putin at the opening of the bridge linking Crimea to southern Russia.
Such a bridge was first proposed under the last Czar Nicholas II before World War I broke out, preventing it going ahead.
After driving a Kamaz truck for nearly 20 minutes, which was part of a column of vehicles that crossed the bridge for the first time, Putin considered that the entire country worked in its construction.
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From the Tsarist era, then in the 1930s and then in the 1950s, an attempt was made to erect a similar work and only you made it possible now, said the Russian statesman to the mass of construction workers who attended the inauguration.
But for others the structure has become a bridge too far. President Petro Poroshenko lambasted the bridge as "yet another piece of evidence of the Kremlin ignoring global law" and "an attempt to legitimise the temporary occupation of the Crimean peninsula".
"This is an extremely important day from this point of view and in a practical sense and in symbolic terms".
The road stretch of the bridge was due to be completed by the end of 2018, but the opening was moved up at Putin's request. A connecting highway from Kerch to the regional capital of Simferopol will only be completed in 2020.
Ships will be able to pass through an arch of the bridge that is 35 metres (115 feet) high in the deepest part of the strait.
The column of trucks, cranes and other heavy-duty vehicles carried construction workers who had been working on one of Russia's most ambitious infrastructure projects for the past several years. It is also created to withstand ice floes in winter.