Hyundai said in a statement that the Estonian, with co-driver Martin Jarveoja, were able to exit the car unassisted after the accident in the French Alps 9.2km into the 20.68km fourth stage.
"Tanak and Järveoja have been taken to hospital for precautionary medical checks, as is standard procedure following a crash of this nature," the team added.
Camera footage showed the Hyundai careering off the road and barrel-rolling into trees.
Tanak won the championship with Toyota last season and the season-opening Monte Carlo rally is his competitive debut with Hyundai, the reigning manufacturers' champions.
The Estonian had been third overall after Thursday's opening night stages. (Reporting by Alan Baldwin, editing by Toby Davis)