The cost of the project is estimated at around 228 billion rubles (about 3.6 billion USA dollars), according to the project's website.
The bridge surpasses the Vasco de Gama bridge in Portugal as the longest in Europe. Many in Russia saw the move as restoring Moscow's rule over a historically Russian region.
Mr Putin, whose approval ratings have remained above 80 per cent since the annexation of Crimea from Ukraine in 2014, personally oversaw the project.
Kerch bridge will definitely need the invaders, when they to flee Crimea, says the Ukrainian President.
The EU has imposed a range of sanctions over Russia's illegal annexation of Crimea in 2014, carried out in retaliation for Ukraine ousting its pro-Russian president in what Russia has condemned as a Western-backed coup.
Crimea is popular among Russians as a summer holiday destination and Putin said in March he would like the bridge to be ready for the season.
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Russia's last tsar, Nicholas II first proposed such a bridge, but the outbreak of World War I prevented 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. "Many didn't believe these plans were possible", Kremlin spokesperson Dmitry Peskov told reporters on Tuesday ahead of the ceremony.
The railway bridge is not yet finished and has a deadline of the end of 2019. Vehicle traffic on the bridge will start on may 16 at 5.30 am. 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 Russian president attended the symbolic opening ceremony on Tuesday, dressed in jeans and a windbreaker, instead of his usual business suit.