Crestor is a statin approved for the treatment of dyslipidaemia, in which disrupted metabolism leads to changes of fats in the blood, and high cholesterol.
As part of the agreement, the German firm will make an initial payment of US$320mln followed by future milestone payments of up to US$30mln.
The divestment is anticipated to close in the first quarter of 2021, the FTSE 100 firm said.
AstraZeneca will continue to manufacture and supply Crestor to Grünenthal during a transition period and will also continue selling the medicine in other countries, including those in North America, in Japan, China and other emerging markets.
"This agreement supports the management of our mature medicines to enable reinvestment into the pipeline and bringing new, innovative treatments to patients,” said Ruud Dobber, Executive Vice President at the pharma giant’s BioPharmaceuticals Business Unit, in a statement.
“Grünenthal previously acquired the rights to several established AstraZeneca medicines and is well placed to ensure continued access to Crestor for patients across Europe."
Shares rose 2% to 7,932p on Tuesday morning.