Emil Forsberg netted the opener for the hosts and also won a 50th-minute penalty that Zlatan Ibrahimovic converted before Nicolai Jorgensen snatched a late goal to leave the tie delicately balanced going into Tuesday's game in Copenhagen.
"You have to be confident," Olsen told a news conference. "I think we have the players who can make the difference, it will be a different game."
A tight, tense affair for much of the first 45 minutes, the game exploded into life on the stroke of halftime when winger Forsberg swept Mikael Lustig's pass beyond Denmark keeper Kasper Schmeichel.
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The crowd had barely taken their seats at the start of the second period when Forsberg was upended by Thomas Kahlenberg after a surging run into the box and Sweden captain Ibrahimovic thundered home the resulting spot kick.
Denmark, who failed to score in their final three group games, created few chances until substitute Jorgensen stole in at the far post to poke the ball home in the 80th minute and throw his side a lifeline.
With the outside of the Friends Arena lit up in the red, white and blue colours of the French flag, a minute's silence for the victims of Friday's Paris attacks was observed before the game.
Both sides also wore black armbands in memory of the 129 people who died.
DIRECT APPROACH
Billed as a Scandinavian clash between rival strikers Ibrahimovic and Nicklas Bendtner, the more direct approach of the Swedes paid dividends while the Danes put on a remarkably flat performance until their late goal.
Ibra's strike partner Marcus Berg wasted a number of early chances although Bendtner could have put Denmark ahead before Forsberg's goal when he steered an effort just wide.
Jorgensen's late goal left the Swedes rattled and Denmark dominated the final 10 minutes without managing to find the equaliser.
"It's always disappointing to concede a goal," said Sweden coach Erik Hamren. "I thought our defence was good for 60 minutes.
"We have won the first game. We know it's only halftime and ... it's going to be exciting on Tuesday."
In Saturday's other playoff first leg, Ukraine beat Slovenia 2-0 in Lviv.
The winners of both ties go through to the finals in France.
(Editing by Tony Jimenez)
