Truss, the fourth Conservative prime minister in six years, flew to the royal family's Scottish home to be asked by Queen Elizabeth to form a government.
She replaces Boris Johnson who was forced out after three years in power.
"We now face severe global headwinds caused by Russia's appalling war in Ukraine and the aftermath of COVID," the 47-year-old former foreign secretary said outside her Downing Street office.
"I am confident that together we can ride out the storm. We can rebuild our economy, and we can become the modern brilliant Britain that I know we can be."
Truss, who will later announce her government appointments, said she had three priorities: growing the economy through tax cuts, dealing with rising energy costs from this week and ensuring people get the care they needed from the state-run National Health Service.
However, she inherits an economy in crisis, with inflation at double digits, the cost of energy soaring and the Bank of England warning of a lengthy recession by the end of this year.
Already, workers across the economy have gone on strike.
Her plan to revive growth through tax cuts while also potentially providing about 100 billion pounds ($A171 billion) for energy has rattled financial markets, prompting investors to dump the pound and government bonds in recent weeks.
Truss has also promised to scrap plans to increase corporation tax on big firms, and to reverse an increase in a payroll tax on workers and employers, designed to raise additional funding for health and social care, with the extra spending coming from general taxation.
Ten-year borrowing costs rose to their highest since 2011 but two-year yields fell, with economists noting that an energy price cap would stop inflation rising in the near term.
"I know that we have what it takes to tackle those challenges. Of course, it won't be easy but we can do it," Truss said.
"I will take action this day and action every day to make it happen. United with our allies, we will stand up for freedom and democracy around the world of recognising that we can't have security at home without having security abroad."
US President Joe Biden was one of the first to congratulate Truss.
The new prime minister will address the latest crises buffeting the UK with a weaker political hand than many of her predecessors.
Having held a place in the cabinet of senior ministers for eight years, she defeated rival Rishi Sunak in a vote of Conservative Party members by a tighter margin than expected, and more of the party's MPs initially backed her rival.
Johnson, who tried to cling on to power in July despite ministers resigning en masse over a series of scandals, told reporters and politicians gathered in Downing Street early on Tuesday that the country must unite.
"It's time for politics to be over, folks," he said in his farewell speech.
"It's time for us all to get behind Liz Truss and her team and her program."
After speaking outside the famous black door, he left London to travel to northeast Scotland and tender his resignation to 96-year-old Queen before Truss followed him into Balmoral Castle to be appointed his successor.
