The error occurred in Scotland at Glasgow's Hampden Park stadium which was hosting one of the day's six matches kicking off the women's tournament and led to a lengthy delay before the game eventually started.
Kicking off almost an hour later than the scheduled start time of 18:45 GMT, North Korea were far more composed on the pitch and went on to record a 2-0 victory.
In the day's other games, reigning Olympic champions the United States, bidding for a third successive gold, came from behind to beat France 4-2, world champions Japan defeated Canada 2-1 and hosts Britain overcame New Zealand 1-0.
"The South Korean flag was shown in the video package on the screen before the kick-off and the North Koreans were naturally very upset about that," Hampden Park media manager Andy Mitchell told Reuters.
"We have made a full apology to the team and the North Korean NOC [National Olympic Committee]. A genuine mistake was made for which we apologise. Steps will be taken to ensure it does not happen again."
"We were angry because our players were introduced as if they were from South Korea, which may affect us greatly as you may know," he said.
"Our team was not going to participate unless the problem was solved perfectly and fortunately some time later, the broadcasting was corrected and shown again live so we made up our mind to participate and go on with the match," he added.
"If this matter cannot be solved, we thought going on was nonsense. Winning the game cannot compensate for that thing."
It was a memorable day for hosts Britain who won the first ever Olympic match they have played, beating New Zealand at Cardiff's Millennium Stadium in the first event of the London Olympics.
Steph Houghton, who missed the 2007 World Cup and 2009 European Championship through injury, scored Britain's winner with a second half free-kick.
