Kagawa stepped up to score the resulting penalty to put Japan in the lead.
The Samurai Blue were in complete control for the next 15 minutes and were almost back in front after Osako got a shot away, which Ospina was equal to on this occasion, as he was when Inui curled one towards the top corner.
Ospina was forced into a pair of saves after the restart, first going down to smother Osako's shot and then diving out fully to palm away an effort from Inui.
Osako created the game's first moment of danger, getting away down the right to provide an early cross that was claimed by goalkeeper David Ospina.
But Japan were dominating possession and their pressure told when substitute Keisuke Honda's corner found Osako, who outjumped his marker and headed the ball into the far corner of the net, beyond Ospina's reach.
He will now try to help Colombia overcome the barrier of being down to 10 men after Carlos Sanchez was sent off at 3 minutes.
Donald Trump to nominate OMB official to lead consumer watchdog
Mick Mulvaney speaks during a news conference after his first day as acting director of the Consumer Financial Protection Bureau. The agency has been the focus of partisan fights since it was created by the Dodd-Frank Act after the 2008 financial crisis.
Colombia hit back! Quintero with the goal just before Half-time!
The ball went out Kagawa and his goal-bound follow-up hit the outstretched arm of Sanchez, drawing an immediate red from referee Damir Skomina.
Yuya Osako had a surprise in store for those who thought Shinji Okazaki's lack of fitness left Japan little hope of competing with Colombia at the World Cup. Also no James Rodrigez for the South American side.
Hello! Welcome to the Live Updates for Colombia Vs Japan. We will have all the latest updates and Video highlights from the game so don't go anywhere! With the "Fubo Premier" bundle, you will get access to all four of the World Cup soccer TV channels in most markets.
But a brilliant free kick has Los Cafeteros level. Providing they can make a strong start, Colombia can realistically target another quarter-final, thanks in part to their kind draw.
Colombia was one of the stories of the last World Cup, with James Rodriguez rising to prominence with his performance in Brazil. The Monaco striker missed out on the 2014 competition with a knee injury and will be relishing the opportunity to represent Colombia on the world stage. Radamel Falcao also had at least one decent chance. Coach Akira Nishino was appointed just two months before the start of the tournament and doesn't have an idea about his first XI and star players are all battling for form ahead of the crucial encounter.
This is the first World Cup tournament in which as many as four South American teams have failed to win their opening game since 1974.