The spy row risks hurting Russia's ties with one of its rare European allies.
It is the latest in a string of cases where Moscow has been accused of espionage in EU states and elsewhere.
She "expressed hope that the steps will not affect the further development of bilateral cooperation," the Russian foreign ministry said in a statement.
Lavrov reiterated Moscow's displeasure that Austria chose to air its grievances in public.
He said the practice of voicing "unfounded accusations" publicly was unacceptable, the foreign ministry said.
"Any possible mutual concerns should be discussed through established channels of dialogue and be based on facts," the statement said.
In August, Kneissl sparked a row in Austria and beyond after she waltzed with Russian President Vladimir Putin and bowed to him at her wedding, with critics saying her behaviour hurt the country's image.
Austrian defence ministry spokesman Col Michael Bauer Bauer told the Kurier newspaper on Saturday that the information passed on by the suspect "ranged from unimportant things to more sensitive information".
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