Kagawa went close in the 16th minute but his shot rolled past the post and Osako fired against the face of goal from a unsafe position in the box.
Despite being a man down the Colombians continued to press and were rewarded with an equalizer six minutes before the break as Quintero sent a free-kick from just outside the area under the leaping Japanese wall. Falcao captain's Colombia in his first World Cup appearance.
An excellent second-half effort from Japan earned them a crucial victory against Colombia, whose hopes were ultimately ruined by the sending off of Carlos Sanchez after three minutes.
Pekerman defended his strategy, which included subbing in attacking midfielder and 2014 Golden Boot victor James Rodriguez, who had been left out of the starting lineup because of muscle pain.
Japan became the first team to beat a South American team with a 2-1 win over Colombia at the Mordovia Arena in Saransk.
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The team says Dzagoev underwent an MRI scan on Friday and "a timeframe for his treatment will be determined in the coming days". They are due to face seven-time African Cup champions Egypt and Uruguay.
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Prime Minister Shinzo Abe told reporters the government was "working united, with its first priority on saving people's lives". Railway services in Osaka, Shiga, Kyoto and Nara, both high speed bullet trains and local rail services, were halted.
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It was the second-fastest red card in World Cup finals history, after Uruguay's Sergio Batista was sent off inside the first minute in 1986.
After Quintero slotted home a clever free-kick against the above 'keeper Kawashima, it resulted in more free-kicks being scored in this World Cup than in the entire Brazil tournament four years ago.
While Carlos Sanchez took the fall, namesake Davinson was equally at fault as he allowed a ball over the top to bounce in the build-up to the penalty incident, giving Osaka a chance to shoot and Kagawa the opportunity to pounce on the follow-up. Colombia brought on James Rodriguez and Carlos Bacca in a desperate bid to salvage a point but Japan held on for a first win at the World Cup since beating Denmark in 2010.
Japan coach Akira Nishino praised his players' ability to adapt and secure the victory through Yuya Osako's 73rd-minute victor.
Colombia continued to rally but the effect of playing with 10 men for nearly the entire game took its toll with Japan holding out, leaving Colombia with a mountain to climb in Group H.