Danish goalkeeper Thomas Sorensen was forced into several fine saves and Italy also had Gianluigi Buffon, the most expensive goalkeeper in world soccer, to thank for keeping them in the Group C match.
"It was very important to get something out of the game. I can only speak for my team but I'm not satisfied with one point," Olsen said.
"In the first half we were in control, Thomas made a couple of good saves at the end of the first half. Then Italy had chances in the second," he told a news conference.
In the final minute before the interval Sorensen made a great double save, first blocking a point-blank shot from Alessandro Del Piero and then making a one-handed parry from Francesco Totti.
"We could have lost if Thomas hadn't made those saves, but overall we deserved to win," Olsen said. "Denmark were better prepared than Italy, we coped with the heat."
Sorensen also brilliantly tipped over a Christian Vieri header in the second half and was named man of the match. In 2002 he made a bad mistake to give England their first goal in a 3-0 defeat in the second round of the World Cup finals.
"All players have bad games but that's well behind us. In the last two years we've been very consistent, even beating big teams like England and Turkey away," Sorensen said. "I'm not surprised we did well today."
Italy coach Giovanni Trapattoni said his team suffered in the heat and he was happy with their second half performance.
"You saw that Italy were in a position to win, yet Denmark did not deserve to lose," Trapattoni said.
"We'll be better physically next match, I'm convinced we must play like we did in the second half. I told the players at halftime to keep (their game) shorter.
"We began to play in the second half but it was very hot," he said. The temperature was measured at 34 degrees Celsius on the pitch in the second half.
"This was a good first test in the tournament, to get a better idea of the reality of the Euro."
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