Colombia's Carlos Sanchez received a red card for blocking Kagawa's shot on goal with his extended right arm.
"Normally you prepare a match to play 11 players against 11 and to lose one player in the first three minutes _ to lose such a crucial player - that's not an easy thing", Colombia coach Jose Pekerman said.
As the half wore on, Los Cafeteros began to grow in confidence and, in the 39th minute, they duly levelled the scores when Quintero cleverly fired a freekick under the Japan wall and Eiji Kawashima was unable to scramble across in time to keep it over the line.
Juan Quintero's clever low free kick in the 39th minute gave Colombia the equaliser but Yuya Osako leapt above his marker to head a corner into the net for Japan's 73rd-minute victor.
Kagawa's penalty kick after just six minutes became Japan's quickest ever goal at a World Cup. The team joins Mexico, Switzerland and Iceland in earning surprising results. Falcao captain's Colombia in his first World Cup appearance.
There have been more direct free-kick goals scored in just 15 games at World Cup 2018 (four) than there were in the entire 2014 edition (three), with Colombia's Juan Quintero netting the fourth of this year's tournament.
Jonathan Evans adapted this story based on the Associated Press news report.
Japan had another excellent chance in the 32nd minute when Osako turned the Colombian defense over, but he sliced his right-footed shot well wide. Goalkeeper David Ospina blocked Osako's shot, but Kagawa pounced on the rebound and quickly got off a shot on goal that drew Sanchez's red card and the penalty.
Colombia came out on top 4-1 when these sides last met in a World Cup group match four years ago, but there only looked like being one victor on the basis of the second half.
"Our players were aggressive from the start and I think that worked very well for us".
Japan will face Senegal in their next game while Colombia will look to redeem their image when they face Poland also in their next game.
With the numerical advantage, Japan did create chances to extend their lead but Takashi Inui and Osako both wasted opportunities from good positions.
Tipped to finish bottom of the group, Japan have now made a blistering start to their World Cup campaign and could reach the Round of 16 for a third time in their history if they build on the momentum from today.
When the game was due to officially start, Falcao walked across to Japan's side of the field and spoke to referee Damir Skomina in a discussion that lasted beyond the official countdown to kickoff.