Having won all of their Group D games, Croatia beat Denmark in a shootout in their last 16 clash to set up a quarter-final against Russia on Saturday in Sochi on the Black Sea coast.
Russia took a similar route to the last eight with a win on penalties over Spain in the Round of 16.
"It'll certainly be a hard game, they play at home, they'll have the fans backing them," Subasic, Croatia's hero of the win against Denmark, told reporters through a translator.
"They're a good team, they play well and of course it'll be hard, as each time so far. Then again, the challenge is that much greater when everything is against you."
Subasic made three saves in their 3-2 shootout win - only the second keeper to do so in World Cup history after Portugal's Ricardo versus England in 2006.
Russia played with five at the back and a lone striker in Artem Dzyuba against Spain and it worked as their opponents managed to create just one chance in 90 minutes on Sunday.
It might not be the same against Croatia, with the Balkan nation having shown flashes of their stellar attacking game when they beat Nigeria, Argentina and Iceland to top their group.
"I don't know, they're playing at home, I suppose they should attack us. I don't know how they'll play, I'm not their coach," Subasic said. "They have a good squad, good individuals... (it will be) a game as difficult as any other.
"It's certain they'll go beyond their abilities, playing at home, in front of their fans. I think we have the answers to all that's waiting for us."
Hrvoje Custic, Subasic's former NK Zadar team mate, died in 2008 aged 24 after hitting his head on a railing near the pitch during a match and the keeper always wears a vest in his memory.
Asked on Tuesday about the message on his vest, Subasic broke into tears. He could take no further questions.
After a patchy qualifying campaign during which Croatia sacked coach Ante Cacic in the closing stages to make way for Zlatko Dalic when they slipped from first to third in the group, they have looked impressive in the World Cup group stages.
"We were in a bad situation. you may say 'in the canal'," defender Domagoj Vida said. "We're aware of our quality, our possibilities and what sort of players we have and, in the end, we were always at our best when it was the hardest.
"We showed in this World Cup that we have the right to hope for big things. We play really well and we have such players that we certainly should believe that we can go to the very end."
(Reporting by Sudipto Ganguly; editing by Ken Ferris)
