This is the first such death since pro-Russia forces took control in February.
Ukraine has now authorized its troops to fire in self-defense.
The attack came shortly after Russian President Vladimir Putin and the leaders of Crimea signed a bill to absorb the peninsula into Russia.
Western powers condemned the treaty and a G7 and EU crisis meeting has been called for next week in The Hague.
The Ukrainian crisis began in November last year after pro-Moscow President Viktor Yanukovych abandoned an EU deal in favour of stronger ties with Russia. He fled Ukraine on February 22 after protests in which more than 80 people were killed.
According to new reports, armed men arrived in two unmarked vehicles, storming the base in Simferopol and firing automatic weapons.
The Ukrainian government said a junior officer who was on duty in a park inside the base had been killed and another officer injured. A third serviceman had leg and head injuries after being beaten with iron bars, it said.
The government said the commander of the unit was captured by men wearing Russian uniforms.
Defense ministry spokesman Vladislav Seleznyov told Reuters the attack was by “unknown forces, fully equipped and their faces covered”.
The Ukrainians had had their IDs, weapons and money confiscated, he said.
Ukraine’s interim PM Arseniy Yatsenyuk told an emergency government meeting: “The conflict is shifting from a political to a military stage.
“Russian soldiers have started shooting at Ukrainian military servicemen and that is a war crime.”
Reports from the Crimean news agency, Kryminform, said a pro-Russia defense force member had been shot dead.
Crimean police later said both Ukrainian and pro-Russian forces had been fired on from a single location and that one Ukrainian was killed and one injured, and one pro-Russian was killed and one injured.
None of the accounts can be independently confirmed.
Until now only warning shots have been fired amid a truce – but it appears the tension has boiled over and there are fears that further clashes could follow.
Earlier, President Vladimir Putin told Russia’s parliament that Crimea had “always been part of Russia” and in signing the treaty he was righting a “historical injustice”.
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