The chamber voted overwhelmingly on Thursday evening in favour of the mandate, 47 to 12, following a vote in the lower chamber of parliament on January 20.
It will now be sent to President Alexander van der Bellen for signing.
"Today is an important day," Health Minister Wolfgang Mueckstein said during the parliamentary debate.
With the vaccine mandate, he said Austria "isn't simply reacting, but we are taking a forward-looking and active step".
Under the law, the first of its kind in Europe, all adults 18 and older will be required to be fully vaccinated against the coronavirus.
Once the mandate goes into effect, authorities will write to every household to inform them of the new rules.
From mid-March, police will start checking people's vaccination status during routine checks. Those who can't produce proof of vaccination will be asked to do so in writing, and will be fined up to 600 euros ($A950) if they don't.
Officials originally announced the mandate would take effect on February 1 but the law still needed to be debated and approved by the upper house.
The news comes just as Austria plans to loosen many of its pandemic-related restrictions. Over the course of February, the country will ease restrictions on unvaccinated people entering shops, restaurants and other areas of public life from which they've been effectively barred since November.
Not all states will follow the national government's lead, however: Vienna Mayor Michael Ludwig announced on Thursday that the capital will continue requiring proof of vaccination or recovery to enter restaurants.
After weeks of rapidly rising cases, Austria's surge fuelled by the Omicron variant of the virus shows some signs of slowing.
The country reported 38,135 new infections on Wednesday, bringing its seven-day case rate down slightly to 2597 per 100,000 inhabitants.
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