Brazilian-born Mario Fernandes had kept Russia's dream alive when he headed home in the 115th minute to level the scores and force penalties, but he and Fedor Smolov both failed to convert their spot-kicks in Sochi.
Croatia, for whom Andrej Kramaric and Domagoj Vida had earlier found the net, advance to their first World Cup semi-final since 1998. They will face England, who beat Sweden 2-0 earlier, at the Luzhniki Stadium Moscow on Wednesday.
In contrast for Russia, who had taken the lead through another Denis Cheryshev wonder goal in the first half, there is only heartbreak as they finally go out.
They can however be proud of their efforts, with a run to the last eight exceeding expectations for a side written off as no-hopers before the tournament. And it remains to be seen how quickly Croatia -- whose President Kolinda Grabar-Kitarovic was watching from the stands in her country's colours -- can recover from this draining night, having also been taken to extra time and penalties by Denmark in the last round.
The Russians, riding a wave of euphoria after unexpectedly knocking out Spain in the previous round, were determined to carry on to a first World Cup semi-final since the break-up of the Soviet Union.
But Croatia came into this game as the favourites. They may have a population of little over four million, but amongst that number are two of the finest midfielders in the world in Rakitic and Luka Modric.
They dominated the match, but Russia contributed to another fantastic occasion in what has been an outstanding World Cup.
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