Sergei Aksyonov wrote on social media that the blasts had taken place at the Saky military base near Novofedorivka on the peninsula's western coast.
Footage circulating on social media appeared to show multiple explosions.
Russia's defence ministry later said ammunition was detonated, but this has not been independently verified.
The ministry said there was no "fire impact" on the ammunition storage area, Russia's state-run Ria news agency reports.
The Ukrainian peninsula that Moscow annexed in 2014 is a popular destination with Russian tourists, and one video appeared to show visitors fleeing from a beach as smoke rose in the background.
Local witnesses told the Reuters agency that they heard at least 12 blasts, beginning at about 15:20 local time (12:20 GMT).
After arriving in the area, Mr Aksyonov said a 5km (three-mile) no-go zone around the blast site had been established.
Moscow refused to speculate on the origin of the blast and Kyiv has not taken responsibility.
Ukraine's deputy Prime Minister, Iryna Vereshchuk, wrote on Telegram that "today's explosions in Novofedorivka are another reminder of who Crimea belongs to. Because it is Ukraine".
And Ukraine's military posted a sarcastic statement on Facebook which hinted at the sinking of the Moskva warship and other military setbacks, reminding Russia of its "fire safety rules and the ban of smoking in unsettled places".
Russia's annexation of Crimea eight years ago drew widespread international condemnation.
On 24 February, Moscow launched a full-scale invasion of Ukraine, using Crimea as a springboard to move Russian troops deeper inside Ukraine.
Novofedorivka and Saky are about 50km (30 miles) north of the port of Sevastopol, home of Russia's Black Sea Fleet, which has been leading a blockade of the Ukrainian coastline.