The teams clash at the Fisht Stadium in Sochi having both reached this stage thanks to penalty shootout victories in the last 16 -- Russia stunned 2010 winners Spain before Croatia edged out Denmark.
Croatia lost 3-1 to Brazil in Sao Paulo in the opening game of the 2014 World Cup. In 1998, they reached the semi-finals before losing 2-1 to France in Paris.
But while they could not claim to be the favourites on either of those occasions, this time Zlatko Dalic's side are strongly fancied Russia, who have surprised everyone by making it this far.
"We cannot choose our opponents, be they the hosts or anyone else," Dalic told reporters on the eve of the game in the Black Sea resort city of Sochi.
"It doesn't matter who our opponents are. Maybe the only thing is the vociferous support of the fans, but every week our players play in front of full houses with rival fans howling, so this shouldn't be a problem, and we shouldn't be looking for any excuses."
For a country of little over four million inhabitants, Croatia appear to have a golden opportunity to reach the last four once again, just as they did in 1998.
While their fans will be massively outnumbered in a crowd of over 40,000, Dalic's side are expected to have one especially notable backer at the Fisht Stadium in the shape of Croatian President Kolinda Grabar-Kitarovic.
"The whole country is euphoric. There are parties going on when Croatia play, and everyone is out in the squares and cafes," said Dalic.
"It was great motvation that our president attended the last game and that she will be there tomorrow (Saturday). We are very happy, and I can only hope she will have the chance to visit us in Moscow."
If Croatia win Saturday's game, they will advance to a semi-final in the Russian capital next Wednesday against either England or Sweden.
Their only previous appearance in the quarter-finals came in their first ever World Cup as an independent nation, in France 20 years ago.
This week marked the 20th anniversary of their stunning 3-0 victory over Germany in Lyon, when Robert Jarni, Goran Vlaovic and Davor Suker got their goals.
Coach Dalic had attended that tournament but missed the Germany game to return to pre-season training with his club.
Ivan Rakitic, meanwhile, recalls watching the match as a 10-year-old in the Swiss town where he grew up.
"I was a child, so I could not go to France, but I remember in my hometown Moehlin we were all going crazy and we wanted to show our Swiss neighbours how powerful a football nation we were," said Rakitic, the 30-year-old Barcelona midfielder.
"They are our heroes, our inspiration, and they brought us to this point, so hopefully we can go a step further tomorrow."
Croatia have a slight doubt as to the fitness of left-back Ivan Strinic and midfielder Mateo Kovacic due to respective calf and shoulder problems.
Meanwhile, there is a question mark as to whether Marcelo Brozovic will keep his place in the starting line-up, with Milan Badelj a contender to come in.