More than a dozen men stormed into the Kiwi cafe in the Georgian capital on Sunday evening, the cafe said, shouting and throwing meat at patrons.
A brawl erupted but the attackers fled before police arrived.
Police are now investigating, and say they have questioned the attackers and cafe staff. Nobody has been arrested.
The cafe has appealed for public support, saying it was no prank but a case of intimidation by neo-Nazis.
The attackers wore strings of sausages round their necks and threw chunks of meat onto customers' plates, the BBC's Rayhan Demytrie reports from Tbilisi.
They are known as the Bergmann group, and a social media page shows their attacks on people of Arab or African origin, our correspondent reports. One photograph shows members making the Nazi salute.
The Kiwi cafe is in a traditional part of old Tbilisi, and is popular among young people sporting unconventional hairstyles, tattoos and body piercings.
Most Georgians are Orthodox Christians and many see unorthodox lifestyles as a corrupting influence from the West.
The cafe said it had drawn some local hostility because of "the way we look, music that we listen to, ideas we support, and the fact that we don't eat meat" and backing of causes such as rights for lesbian, gay, bisexual and transgender (LGBT) people.
"During these hard times you can support us just by visiting our cafe, we will be very grateful if you come to show everyone that here are a lot of us who care about the issue!" the Kiwi cafe said in a statement on Facebook.
It later told the BBC it had received strong messages of solidarity - from Georgia and abroad - but that people had also left angry comments trying "to defend those fascists".
It said it was also appealing for financial contributions to install security cameras in case of repeat incidents.
The incident comes amid growing concerns about the rise of far-right nationalism in Georgia.
Last week, hundreds of nationalists marched through central Tbilisi - waving Georgian flags and anti-communist banners, reports said - to mark independence from the Soviet Union in 1991.
Homophobia is also commonplace in Georgia, correspondents say. The country made world headlines in 2013 when a small group of LGBT activists were attacked by a large mob led by an orthodox priest.
The cafe statement blames Sunday's incident on the group of men who came into the premises, talking loudly, throwing meat and smoking, and then "yelling, laughing, and talking to us sarcastically".
These people "were neo-Nazis... who support fascist ideas", the cafe said.
Some minor injuries were sustained.
The police arrived only after the attackers had left, but the cafe said even some of those officers behaved aggressively, "yelled with anger, said that we are guilty of what had happened".

The proposal, dubbed Hornsea Three, would create a complex that would be the "world's largest offshore wind farm", Dong Energy said.
Hornsea Project One, which is being built 75 miles (120 km) out at sea, is expected to be operational in 2020.
The RSPB has previously voiced concerns about the "high collision risk for seabirds using the area".
Brent Cheshire, Dong Energy UK's chairman, said moving forward with Hornsea Project Three "underlines our commitment to the UK offshore wind industry."
An application for consent to build Hornsea Project Two was submitted to the Planning Inspectorate in January last year. A decision is expected later in the year.
The energy firm said it expected to submit a planning application for Hornsea Project Three in 2018. If approved, construction was anticipated for sometime between 2022 and 2025.
Dong Energy has already built the Westermost Rough wind farm which consists of 35 turbines located 16 miles (25km) off the Holderness coast.
A vegan cafe in Tbilisi has appealed for public solidarity after being invaded by ultra-nationalists wielding grilled meat and sausages.
[NEXT_CONCEPT]
A public consultation into plans for a third offshore wind farm off the East Yorkshire coast has started.