The men are likely to be liable to pay a large sum in damages after crashing into the monument at the end of a no-through road, reports news agency Ritzau.
“They continued to drive over the lawn and into the monument,” duty officer Erik Lindholdt of Southern Jutland Police told the agency.
Lindholdt said that his guess was that the men thought they could take a short cut across the lawn.
The memorial consists of a bronze relief with a row of vertical stones with engraved names.
“Many of the stones are lying in pieces at the scene,” Lindholdt told Ritzau.
Following the accident, the two men refused to tell police which one of them had been behind the wheel of the car, so both were arrested and charged with driving under the influence of narcotics, reports the news agency.
Driving under the influence of cannabis or other illegal substances in Denmark and be punished with a driving ban of up to three years.
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Blood tests will be used to confirm whether the two men were intoxicated while driving the car.
The now-partially destructed monument is located at Fovrfeld Gravlund cemetery in Esbjerg - Denmark's largest war cemetery, according to the city's tourist office website.
The monument itself lists the names of Esbjerg citizens that lost their lives to the 1939-45 conflict.
1,150 German soldiers and 152 German refugees, as well as 271 Commonwealth airmen from the Second World War are also buried at the site.