Japan are the first Asian side to beat a South American side at a World Cup.
In the third minute, Osako won a bouncing ball from defender Davinson Sanchez and broke in alone toward goal.
In the early minutes of the game, there was controversial drama as Carlos Sanchez was handed the first red card of the tournament for stopping a goal-bound shot of Shinji Kagawa with a raised arm in the box.
A goal and a man down, Colombia faced an almighty task to fight their way back into the contest.
They evened the score late in the first half on a rolling free kick by Juan Quintero. With Japan rotating between the Werder Bremen forward, Mainz's Yoshinori Muto and Leicester's Shinji Okazaki as the lone target man in their warm-up matches, Osako's 73rd-minute victor showed he was the man for the job.
Akira Nishino was only appointed as Japan coach in April, replacing Vahid Halilhodzic during a tumultuous build-up to their World Cup campaign.
Not just the second-fastest goal of the tournament, the game which squared off on a dramatic note, also saw the second fastest red card in World Cup history.
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.
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Jonathan Evans adapted this story based on the Associated Press news report.
Kagawa converted the resultant penalty with a right footed shot to the centre of the goal to give Japan the lead.
Japan coach Akira Nishino praised his players' ability to adapt and secure the victory through Yuya Osako's 73rd-minute victor. Sanchez did his best to get in the way of the ball. "This is World Cup, it's a huge stage, and I think we started off very well".
Brazil 2014 Golden Boot victor James Rodriguez came on for the last half-hour after labouring in training with a calf strain but could not inspire his team to pull back on level terms. "This is just one win and three points, so we will save our celebration".
Ospina did well to keep out a curling effort from Inui at full stretch three minutes before the hour mark, while Hiroki Sakai had a low shot desperately blocked in the 73rd minute after good hold-up play from Osako.
The result was bitterly disappointing for Colombia coach Jose Pekerman, with Robert Lewandowski-led Poland, then Senegal to come in Group H.
Nishino responded by dispatching his own talisman, Honda, from the bench a few minutes later and it was the former AC Milan man who impacted the game first. "I wanted to play along the sides so that they could pass balls forward so that we could cause Japan more problems", said Pekerman.
World Cup games are known for starting on time.