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Centre-back Samuel Umtiti's 51st minute header from a corner gave France a 1-0 win, as they defended deep to close it out and reach a third World Cup final.
Samuel Umtiti headed in a 51st minute winner to send Les Bleus through to their third final in 20 years.
Playing in a 'false nine' role, Messi capped a disappointing tournament with another subdued display at Kazan as Argentina bowed out of the last 16 with a 4-3 defeat.
In the week of his 31st birthday, Messi was again a largely anonymous figure as they lost 4-3 to France in a thrilling last-16 match in Kazan.
Mexico have won six of their last nine competitive fixtures against Brazil, losing just twice.
The two teams come into the knockout phase off the back of very different performances.
Russia keeper Igor Akinfeev saved Koke's and Iago Aspas' spot kicks in their first ever penalty shootout following a 1-1 draw over 120 minutes.
Having scored just two goals in navigating their way out of Group C as runners-up, the Danes expect to open up on Sunday.
Ivan Rakitic stroked home the winning penalty to send Croatia into a semifinal since 1998 after a night of unremitting tension in Sochi.
A fourth-minute goal from Thomas Meunier and an 82nd minute Eden Hazard strike earned Belgium the victory and the third-place.
Results 131-140 of 179
