Ukrainian First Deputy Prime Minister Arbuzov will serve as acting prime minister, Interfax Ukraine reported.
Azarov offered his resignation to President Viktor Yanukovych earlier Tuesday in an attempt to help end the deadly violence that has gripped the former Soviet state.
"I have made my personal decision to ask the Ukrainian president to accept my resignation from the post of Ukrainian prime minister in order to create additional opportunities for a socio-political compromise and settle the conflict peacefully," Azarov said in a statement.
Ukraine's Parliament also repealed nine of 12 restrictive anti-protest laws Tuesday soon after Prime Minister Mykola Azarov offered his resignation.
During the special session of Parliament, 361 lawmakers voted to repeal the laws passed Jan. 16, TV Novosti reported. The laws' repeal had been a key demand of anti-government protesters who have been demonstrating against Yanukovych's government since November.
Under the Ukrainian Constitution, the resignation of the prime minister means the resignation of the entire Cabinet, Interfax Ukraine said. However, the Cabinet can still perform its functions until a new government is formed, but for no longer than 60 days.
Azarov spokesman Vitaliy Lukyanenko told Interfax-Ukraine Yanukovych instructed the Cabinet to "continue performing its duties before a new government can start its work. This means that, in keeping with the law on the Cabinet of Ministers of Ukraine, the first deputy prime minister will perform the prime ministerial duties."
The Interior Ministry said three protesters stabbed and wounded three police officers in Kherson and that a police officer later died, the BBC said.
With the police officer, the number of people killed in violence linked to the protests is at least five.
Ukraine's parliament, the Verkhovna Rada, also adopted without debate a package of four bills that would replace the repealed Jan. 16 laws, shoved through by members of Yanukovych's Party of Regions, Interfax-Ukraine reported.
"One question that we have finally agreed upon is the repeal of the laws of Jan. 16," Svoboda party leader Oleh Tiahnybok said. "Nine of the 12 laws voted for ... will be canceled."
Members of Parliament voted to reapprove measures that would amend the country's Tax Code, outline how amendments are introduced for certain laws, amend the Criminal Code concerning crimes committed by Nazis and their followers, and institute penalties for desecration of graves of World War II fighters.
Justice Minister Olena Lukash said Monday opposition leader Arseniy Yatsenyuk, head of the Fatherland Party, had refused Yanukovych's offer to be prime minister. If he had accepted, he could have dismissed the current government.
Yanukovych made the concessions during discussions Monday with the three main opposition leaders: Yatsenyuk, Tyahnybok and Punch Party head Vitali Klitschko.
Lukash confirmed late Monday the Draconian anti-protest laws would be repealed and protesters who occupied the Justice Ministry building would be granted amnesty, provided they vacated "all seized premises and roads," the BBC said.
Anti-government demonstrators took over the Justice Ministry building Sunday, but left Monday after Lukash threatened to impose a state of emergency.
The European Union's foreign policy chief, Catherine Ashton, expected to arrive in Kiev Tuesday to try to defuse the crisis, also had urged the government not to impose a state of emergency, saying in a statement such a move "would trigger a further downward spiral for Ukraine, which would benefit no one.
U.S. Vice President Joe Biden called Yanukovych Monday night to repeat U.S. support for "a peaceful, political solution to the crisis," the White House said.
The clashes since the anti-demonstration laws were passed are an escalation of public protests prompted by Yanukovych's about-face in November concerning a planned trade deal with the European Union and turn toward Russia instead.