The six-time world champion started second, but managed to finish the race at the Nurburgring in first position.
The second was the Dutch pilot of Red Bull Max Verstappen.
The three prize-winners were closed by Australian Daniel Riccardo from Renault.
Finn Valtteri Bottas, who started from pole position, left the race in Mercedes for technical reasons.
Russian Alpha Tauri pilot Daniil Kvyat started 13th, but finished the race in 15th.
During the race, he had a collision with Red Bull racers Alexander Albon, as a result of which the Russian car lost its front wing.
Your full classification after a memorable race at the 'ring!
NINE teams score points!
Including the first of the season for Grosjean and Giovinazzi 👏 # EifelGP 🇩🇪 # F1pic.twitter.com / 9Ajcuwx9K4
- Formula 1 (@ F1) October 11, 2020
Earlier it was reported that Red Bull was told when they would decide on a new supplier of motors.