Tunisia, encouraged by a bigger crowd of supporters, who booed the Danes every time they touched the ball, was close to opening the account for several times.
Anis Slimane made a brilliant solo run in the 23rd minute before beating Kasper Schmeichel in the box, but his goal was disallowed for offside.
They should have pulled ahead in the 43rd minute when Issam Jebali raced to the box with only Schmeichel to beat, but the former Leicester goalkeeper outstretched his arms to keep the shot out.
Christian Eriksen, playing his third World Cup, proved that he was still the team's soul after recovering from the collapse on the pitch following a cardiac arrest at Euro 2020. He caused trouble constantly to the Tunisia defense with set pieces.
The 30-year-old Manchester United star created the best chance as the Danes started to play more vigorously in the second half. He delivered a powerful shot from the edge of the box but his effort was saved by Tunisia keeper Aymen Dahmen.
Tunisia survived a penalty scare in injury time as the ball hit the arm of Yassine Meriah. Referee Cesar Arturo Ramos waved off Denmark's plea after checking the VAR.
It was the first goalless draw after six matches at the World Cup. Tunisian fans exploded in celebration after the final whistle. They have enough reasons to celebrate by scoring a precious point against the 10th-ranked Denmark, 20 places above them.
Tunisia will next play Australia, and Denmark will take on reigning champion France.