Allawi acknowledged the protesters in a video posted on Twitter after his appointment was announced on Saturday, reports Efe news.
"My power is derived from you. If it were not for your courage and sacrifices then there would have been no changes in the country," he said.
"You protested for your homeland, and if I am not able to fulfil your demands that I am unworthy of this position."
The 65-year-old Allawi held the post of Communications Minister in the government of Nouri al-Maliki before quitting in 2012 over what he described as "political interference" by the then-Prime Minister.
"I'm an employee carrying your trust, so do not go back until you get what you want, whether from me or someone else," he said, encouraging protesters to remain in the streets.
He pledged to pay compensation to the families of the 467 protest "martyrs" and to ensure proper medical care for the more than 9,000 others who have been injured.
Though his chief responsibility will be to set a date for new elections, Allawi likewise vowed to work to improve the economy and battle corruption.
The new Prime Minister, who has 30 days to form a government that must then secure parliamentary approval, said that he would brook no attempts by the various political factions to control his choice of Cabinet ministers.
"If the political blocs try to impose their candidates on me, then I will resign," Allawi said.
His predecessor, Adel Abdul-Mahdi, resigned on December 1, 2019 in a bid to mollify the protesters, but had remained in a caretaker capacity given the inability of lawmakers to agree on a successor.
The parties finally settled on Allawi, but only after Saleh gave Parliament until February 1 to make a decision or see him assert his constitutional prerogative to act unilaterally.
On Friday, the spiritual leader of Iraq's Shia Muslim majority, Ayatollah Ali al-Sistani, urged politic to expedite the formation of a new government.
The protest movement that began last October entered a new phase following the January 3 killing of Iranian Major General Qasem Soleimani in an American drone attack in Baghdad.
Meanwhile, Moqtada al-Sadr, an influential Shia cleric and militia leader who had supported the protests, withdrew his backing to focus on the push to get the US to leave, though he renewed contacts with the protesters on Friday.
He also welcomed the appointment of Allawi, saying: "We hope that Allawi does not bow to external and internal pressures, that he announces his program and expedites early elections... and that he seeks the sovereignty and independence of Iraq."
( With inputs from IANS )