The 62-year-old had been due to step aside after next year's European Championships in Poland and Ukraine, which the Danes qualified for by winning their group ahead of Portugal and Norway.
"The decisive point was that I felt a new support," Olsen told a packed news conference at Danish FA headquarters.
"Players have tapped me on the shoulder and told me that it was a bad idea for me to quit. They told me I should continue."
As soon as qualification for Euro 2012 was assured, Danish FA general secretary Jim Stjerne Hansen declared that he wanted to see Olsen continue in the job.
"Much of our success is down to Morten," Hansen told Reuters earlier this month. "He has a lot to say about the way we play, from under-17, under-19 and under-21, all the way through to the A team. We are working with his philosophy."
Hansen said he had offered Olsen the job before the final two Euro qualifiers against Cyprus and Portugal.
"We met on the Sunday evening before the meetings with Cyprus and Portugal. I could see the old boy wanted to stay on, so I offered him the job because I felt strongly that I supported him," Hansen added.
Olsen is one of the longest-serving national team coaches in world football having taken over as Denmark coach in June 2000. Since then, he has led his side to the finals of four major championships.
He spent much of his playing career in Belgium where he was famed for his skill and vision, and he won 102 caps for Denmark before retiring as a player in 1989.
His coaching career started at Copenhagen club Brondby in 1990 before moving on to Cologne and Ajax, whose 4-3-3 style is widely appreciated and imitated in Danish football.
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