Their attempt came just hours before four crossing points along Finland's long border with Russia were due to be closed.
Helsinki wants stop a recent flow of Middle Eastern and African migrants to its border which it says Moscow is encouraging.
Around 300 asylum seekers have arrived in the past week, border guards say.
After Ukraine was invaded Finland joined NATO in April which greatly angered Russia.
For years Finland had been non-aligned but changed its stance after Russia invaded Ukraine.
Finnish Prime Minister Petteri Orpo and Interior Minister Mari Rantanen ordered the southeastern crossing points -- Imatra, Niirala, Nuijamaa and Vaalimaa -- to be closed.
There are currently nine crossing points, with one dedicated to rail travel only.
The border runs a total of 1,340 kilometres mostly through thick forests in the south, all the way to the rugged landscape in the Arctic north.