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."
"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."
"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.
