"Our aim is to be in the final," Advocaat, who will quit his job to take charge of Dutch club PSV Eindhoven after Euro 2012, told a news conference in Moscow on Monday.
"I had recently signed a contract with PSV but right now I'm only thinking of my job with Russia.
"When people think of Euro favourites they usually name Germany, Netherlands and Spain. But don't count out Russia.
"If we play to our full potential, we could surprise a few people. I would call us a dark horse of the tournament. If you recall Euro 2004, nobody expected Greece [to win] it."
The Dutchman said he was disappointed to leave Russia after choosing not to extend his current deal with the Russian FA.
"I'm sad to say goodbye to Russia but I had no choice when the FA bosses said they would offer me a new deal only after the Euros," he said. "I just couldn't agree to such an offer. I'm not a young coach who hasn't achieved anything yet."
Advocaat said Arsenal forward Andrei Arshavin, currently on loan to Russian champions Zenit St Petersburg, would remain Russia skipper and explained why he has dropped Diniyar Bilyaletdinov from a preliminary 26-man squad for Euro 2012.
"I was disappointed with his recent form," he said of the former Everton midfielder, who joined Spartak Moscow in January.
"He didn't play very much at Everton, then didn't do well after joining Spartak. So I didn't think twice picking a left winger between him, Arshavin or [Alexander] Kokorin."
Euro 2012, co-hosted by Poland and Ukraine, starts on June 8 with Russia playing in Group A along with Poland, 2004 European champions Greece and the Czech Republic.
