The tunnel, which runs beneath the Alps, will remain closed for several hours while the wreckage is cleared.
Investigators said it appeared that the driver of the German-registered car had crossed the central line.
The 10 mile (17km) tunnel, which opened in 1980, is a main north-to-south route for Europe and is used by more than a million freight lorries a year.
In 2001, 11 people were killed when two lorries collided head-on in the tunnel, triggering a huge fire. It reopened with stricter safety measures in place.
Last year, the Gotthard rail tunnel was opened to try to relieve some of the traffic using other routes.
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