Osako's 73rd-minute winner meant the Japanese avenged their 4-1 mauling in the group stages of Brazil 2014 as Colombia played with a man down for 86 minutes in Saransk.
After leaving Brazil without a win four years ago, Japan made a dream start to their Russian campaign even though head coach Akira Nishino was only appointed in April.
In an explosive start to the Group H clash, Colombia defender Carlos Sanchez earned the first red card of Russia 2018 with a handball after just four minutes.
Japan took a shock lead when Shinji Kagawa netted the resulting penalty before Juan Quintero equalised with a free-kick for Colombia to make it 1-1 at half-time.
Brazil 2014 topscorer James Rodriguez came on for the last half-hour after labouring in training with a calf strain but could not pull his side level after Osako's goal.
Travelling Colombia fans turned the Mordovia Arena into a sea of yellow, but were soon stunned into silence.
When Osako fired in a shot from Japan's first attack, Sanchez blocked the effort with a raised arm.
Referee Damir Skomina showed him a straight red card after pointing to the spot without referring to the Video Assistant Referee (VAR).
The Colombians bitterly protested but Kagawa drilled home the spot kick to put Japan ahead in the sixth minute.
It was the second-fastest red card in World Cup finals history, bettered only by the 52 seconds it took Jose Batista of Uruguay to be sent off against Scotland at Mexico '86.
After the dismissal, Colombia poured forward and their captain Radamel Falcao twice went close.
Colombia coach Jose Pekerman made a tactical switch on 31 minutes, with Wilmar Barrios replacing Juventus midfielder Juan Cuadrado.
The pressure paid off as Quintero's low free-kick flew under Japan's wall and crept inside the post shortly before half-time. The goal was confirmed by goal-line technology.
Japan pressed after the break, forcing Colombia goalkeeper David Ospina into a string of saves.
To a deafening roar from Colombia fans, Rodriguez came on for Quintero on 58 minutes, just before Barrios earned a yellow card for clattering Kagawa from behind.
Japanese pressure paid off when Osako, who was a constant menace to the Colombia defence, leapt highest from a corner and guided his header in off the post with 17 minutes to go.
The goal jolted Colombia into life as Rodriguez then Barrios went close at the other end.
Bayern Munich star Rodriguez earned a late yellow card for sliding into Japan midfielder Genki Haraguchi.