The River Narva (file image), with Estonia on the left and Russia on the right, flows into the Baltic Sea. Getty Images
Cubans are said to have told guards they were hoping to get to Spain, where one of them has a daughter living.
Cuban citizens only need their passports to visit Russia for tourism but cannot enter the EU legally without a visa.
The three people caught on the River Narva in Russia were placed in a holding centre for migrants near the city of St Petersburg, and given a fine and deportation order.
They had planned their crossing after carefully studying the local terrain and getting the air beds, the guards said.
Bags containing personal belongings, toiletries and identity papers were found with them.
Migrants have occasionally tried to cross the Narva from Russia before. In 2018, a man claiming to be an Egyptian was rescued from drowning by Estonian border guards in 2018 when he tried to swim over.