The Danes finished second behind Italy in Group B, but missed out due to having the lowest points tally of all the nations who finished as runners-up.
Olsen, who has been in charge since 2000, said that he wants to get the team back on track, starting with qualification to the Euro 2016 finals in France.
"After finishing top in the two previous qualification rounds for major tournaments, there are probably a lot of people who are annoyed with only finishing second this time and not being part of the World Cup next summer," the 64-year-old told the DBU's official website.
"Now we are putting that disappointment behind us and we will use the experience in a constructive manner.
"My ambition is to get the national team back on track, for which the players have the potential, and to make the team the national rallying point, which follows from good results."
The draw for Euro 2016 qualifying takes place in Paris on February 23.