The police officer died in hospital, where he had been taken following a heart attack when the regional Basque Country police force Ertzaintza was trying to stop street battles between supporters of the two teams, the police said.
At least five people were arrested.
Spanish Prime Minister Mariano Rajoy offered his condolences and condemned the violence on his official Twitter account.
"My condolences to the family and friends of the policeman who died in Bilbao and my consideration to the Ertzaintza for its efforts to protect those who do enjoy sport. I condemn in the most energetic way the violent events."
Spartak won the second leg of the round of 32 tie 2-1 but were knocked out of Europe's second-tier club competition 4-3 on aggregate after losing the first leg 3-1 in Moscow.
Russia hosts the World Cup this year and local authorities are putting strict security procedures in place to control hooligans, including Russian fans who caused trouble at the 2016 European Championship in France and other soccer events.
(Reporting by Julien Toyer; editing by Ken Ferris)
