The Danes created several chances but lacked bite without striker Nicklas Bendtner who was serving a one-match ban for revealing a betting company logo on his underpants at Euro 2012.
The home side's best chance of the opening period fell to Michael Krohn-Dehli, who fired wide after chesting down a cross from the right in the Group B encounter.
Moments later, Dennis Rommedahl nodded wide as he tried to loop his header from Krohn-Dehli's pass over keeper Petr Cech.
Denmark continued to press in the second half, with seasoned winger Rommedahl tormenting the Czechs with a succession of dribbles, crosses and shots.
Cech had to be on his toes to keep out a Christian Eriksen free kick, and from the resulting corner Andreas Cornelius almost marked his Denmark debut with a goal.
The Danes were denied a late penalty when fullback Daniel Wass was bundled over in the area but German referee Wolfgang Stark waved away their appeals.
In Friday's opening Group B games, Armenia won 1-0 in Malta, while Italy and Bulgaria played out a 2-2 draw in Sofia.
- Dennis Rommedahl
- Nicklas Bendtner