"I'm the top administrator and all of the political responsibility rests on my shoulders, so I offered my resignation to the president, and he agreed," Liu told a news conference.
Wu Den-yih, 61, the secretary general of the Nationalist Party, has been named by the president to replace Liu.
Wu's nomination does not require parliamentary approval, and he is expected to name a new cabinet in a few days after discussing the nominees with the president.
"We will unite and strive with our best efforts to shoulder the difficult task ahead," Wu said.
Wu is a veteran parliamentarian who previously served eight years as mayor of Kaohsiung, Taiwan's second-largest city.
Paul Chiu, Taiwan's low-profile vice-premier, also resigned on Monday.
Chu Li-lun, a county magistrate, was named to replace him, the president's office said.
Morakot caused massive landslides that wiped out whole sections of villages in the island's southern mountains.
While public anger over the handling of the disaster has been the toughest test for Ma since he became president, analysts said they expected no major fallout.
Daniel Soh, an economist at Forecast Limited in Singapore, said: "Taiwan is politically similar to South Korea where real power sits with the president.
"There is unlikely to be much political instability as a result of the premier's resignation."
The cabinet has earmarked $3bn for reconstruction work and will build 1,000 homes for the storm survivors with the help of a charity group and electronics maker, Hon Hai Precision Industry Co, said Ma.