Grohe saved from Martin Benitez after the previous nine kicks had been scored to give the Brazilian side the 5-4 win on penalties following 120 minutes of a tense encounter in Porto Alegre.
The two clubs drew the first leg 1-1 last week in Argentina.
"I got down to save it but this was a victory for us all," Grohe said. "This one writes our name in history."
The Supercup is contested between the winners of South America's two main club competitions, the Copa Libertadores, won by Gremio, and the Copa Sudamericana, won by Independiente.
It was the second time Gremio have won the trophy and comes 22 years after their first triumph, which was also against Independiente.
Gremio had the best chances over the first 90 minutes with Everton coming close on two occasions and Martin Campana saving well from Luan.
The main talking point, however, was late in the first half when after a consultation with the video referee, the Paraguayan referee sent off Independiente defender Fernando Amorebieta for a bad challenge on Luan.
Independiente, however, hung on and could have scored twice in extra time but their forwards could not get touches on crosses driven across the face of the goal.
(Reporting by Andrew Downie; Editing by Greg Stutchbury)
