The Spanish national police also confirmed the officer's death in a tweet.
In a statement on Thursday, police in Germany said they detained "the wanted man" on his arrival in Erdinger Moos from Moscow.
Bilbao had been calm until around 19.45 hours when the most violent of the Spartak "fans" provoked a series of incidents close to the San Mames Stadium throwing bottles, golf balls and fireworks and setting off flares.
The fighting spread onto the streets near the stadium and police struggled to restore order.
After the game, Athletic Bilbao, tweeted: "Our heartfelt condolences to the family and companions of the deceased officer today".
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"Five people have been arrested".
"The Athletic Club shows its strongest rejection of the violence generated around football".
"We can confirm that a Basque policeman died Thursday", said a press spokesman for the Ertzaintza said. "I don't know if would have been better or worse to suspend the match". Spartak posted a lengthy call for good behaviour on their official website, reminding supporters that there were no tickets available and that "the cashier's office will not work".
Athletic said that security would be reinforced in anticipation of "the large influx of visiting fans".