The appointment follows a week of upheaval in the island nation, which has seen the forced resignation of their conservative government over the economic crisis.
Sigurdardottir, a Social Democrat, is heading an interim minority coalition including the Left Green party and two independent ministers.
The coalition will rule until elections on April 25.
'Laissez-faire policy out'
Sigurdardottir, who held a meeting with her cabinet and Grimsson on Sunday evening, said one of her first acts "will be to change the leadership of the central bank".
For the first time the cabinet's 10 posts are evenly split between men and women.
Steingrimur Sigfusson, the Left Green party leader, has been made finance minister. The appointment marks a move to the left after right-wing rule.
"Today, laissez-faire economic policy leaves Iceland, which is severely wounded after years of it running the government," Sigfusson said.
The new government said in a statement that it would conduct "a prudent fiscal policy" and protect the welfare state.
Sigurdardottir, 66, is a former flight attendant and union organiser, and served as social affairs minister in the previous government.
She is the second openly gay national leader of modern times, after Per-Kristian Foss, who was briefly Norwegian prime minister in 2002.
Thousands of Iceland's 320,000 citizens have protested in recent months against their country's leadership. The country had been one of the richest in the world, until its banking system collapsed in the autumn due primarily to heavy debts.
Iceland's currency has collapsed, unemployment is expanding rapidly and thousands of people have lost their savings, forcing it to take $10bn in loans from the International Monetary Fund and other countries.