The 34-year-old, driving an Aston Martin, spun into the barriers at Tertre Rouge on lap three after 10 minutes.
"In a serious condition, Allan Simonsen was transferred immediately to the Circuit Medical Centre where he died soon after due to his injuries," read a statement on the event's website.
Simonsen, widely regarded as an extremely fast endurance driver, was competing in the GTE Am class in the 90th anniversary edition of the annual epic sportscar race.
Media reports said he was conscious and talking to doctors immediately following the accident.
He was attended at the scene of the crash by staff from the Automobile Club de l'Ouest's medical service who transferred him to the Circuit Medical Centre where he died.
Aston Martin team chief David Richards informed his family, who said they wanted the team to carry on in the event in tribute to Simonsen.
Aston Martin Racing Managing Director John Gaw added in a statement: "On behalf of all of us at Aston Martin Racing, I would like to extend our deepest sympathies and condolences to the individuals, and families whose friends or loved ones were involved in today's terrible tragedy."
Simonsen had taken part in seven Le Mans 24 Hours events.
His co-drivers in the car, numbered 95, were compatriots Christoffer Nygaard and Kristian Poulsen.
In a statement, the organisers (Automobile Club de l'Ouest) said that no further information will be released "while the exact reasons for the accident are still being determined".
Simonsen was quoted telling reporters on Friday that he was looking forward to the race with great anticipation.
"Just being at Le Mans and to be allowed to drive the car on the track is incredibly special," he said. "It's the biggest race for us.
"It's the one we most look forward to. It's also difficult because there are so many excellent drivers from all over the world - it's a huge challenge."
After Simonsen's accident, the first hour of the race was led by the safety car until waved green flags signalled renewed racing.
Three hours after the start, German Andre Lotterer led the race in the number one Audi R18 e-tron quattro diesel hybrid ahead of two other Audis.
Simonsen's death is the first driver fatality at Le Mans since Frenchman Sebastien Enjolras was killed during pre-qualifying in 1997 and the first during the race since Austrian Jo Gartner died in 1986.
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- Allan Simonsen
- Aston Martin Racing