Victory over the hosts in Sochi on Saturday would put Croatia, led by the wispy and shimmering midfielder Luka Modric, close to their best ever showing -- third in 1998.
Yet how they struggled to get past dogged Denmark in Sunday's slightly surreal last 16 game in Nizhny Novgorod.
In an extraordinary opening, both teams scored comical goals in the first four minutes. But instead of that setting the stage for a thriller, the rest of normal and extra-time was mostly poor quality, as Denmark successfully stifled Croatia's creativity and posed their own constant threat from long throws.
On both sides, passes were mistimed, crosses over-hit, and shots scuffed.
Modric, so devastating in the group stage, hung oddly deep and looked lethargic - until just before the end when he showed his class with a perfect long pass that drew a foul in the area from Danish keeper Kasper Schmeichel.
The 32-year-old, now one of the biggest names left in Russia given the early exits of Lionel Messi and Cristiano Ronaldo, hit a tame shot that Schmeichel saved, and punched the air in anger minutes later as the whistle went for a dreaded penalty shootout.
Nevertheless, Modric bravely returned to take - and convert - a spot kick, while goalkeeper Danijel Subasic saved three from Denmark. Croatia's 3-2 shootout win saw fireworks, flares and roars rise from crowds of fans back home in Zagreb.
"We haven't played a great game ... Without luck, you cannot do anything in life," Croatia coach Zlatko Dalic told reporters, after he had calmed down from celebrating with fans and players.
Russia, who also won a penalty shootout against Spain on Sunday after keeping them at bay with an energy-sapping rearguard performance, will have taken careful note of how those same defensive qualities frustrated Croatia against Denmark.
They, too, will look to shackle the likes of Modric and fellow midfielder Ivan Rakitic, and then hit on the counter as they have done with such surprising success at the tournament.
Dalic said once the euphoria of reaching the quarter-final was over, he would tell his team to avoid arrogance and prepare as if they were playing "the best team in the World Cup".
But in truth he will know Russia are an inferior side, and his team have a fantastic chance of emulating the famous 'Class of 1998' who came third in France.
(Editing by Amlan Chakraborty)
