Putin, at the wheel of a heavy KAMAZ truck, drove the full 19 km across the bridge from the Kerch Strait, which some Russians call "Putin's bridge" and which is created to link Crimea into Russia's transport network.
The bridge was completed six months ahead of schedule.
Ukrainian Prime Minister Volodymyr Groysman condemned the project, saying it has damaged the environment and that larger ships will not be able to pass through its ports on the Azov Sea.
European Union and United States 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.
Construction on the bridge to link Crimea with Russia's Krasnodar region began in February 2016.
Opening the bridge across the Kerch Strait is in fact an attempt to legitimize the annexation of part of Ukraine's sovereign territory against the backdrop of Russia's armed aggression against Ukraine and the temporary occupation of the Crimean peninsula, the ministry said.
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Such a bridge was first proposed under the last Czar Nicholas II before World War I broke out, preventing it going ahead.
The bridge is set to be opened to cars on Wednesday.
"This constitutes another violation of Ukraine's sovereignty and territorial integrity by Russian Federation", said a statement by a spokesperson for the office of the European Union's foreign policy service, which said the bloc continues to refuse to recognise Russia's annexation of Crimea.
The Ukrainian Foreign Ministry stressed that Ukraine would consistently work to expand national and global sanctions against individuals and legal entities involved in the illegal construction of the Kerch bridge.
Originally, the bridge was expected to be built before 2018 and put into operation in the summer of 2019.