The first concrete concessions by Mr Yanukovich since the crisis erupted two months ago brought cheers of support from several thousand demonstrators on Kiev's Independence Square, the focal point of the ongoing protests.
Opposition leaders said they would continue to harness street power to wring yet more gains from Mr Yanukovich. Boxer-turned-politician Vitaly Klitschko said: "We have to change not only the government but the rules of the game as well. We are sure the struggle will continue."
Mr Azarov, 66, announced his resignation as parliament met for an emergency session to flesh out a deal that would satisfy the opposition and end violent street protests in the capital Kiev and other cities.
Mr Azarov, a loyal lieutenant to Mr Yanukovich since the latter was elected to power in February 2010, said he was offering to step down "with the aim of creating extra means for finding a social-political compromise, for the sake of a peaceful settlement of the conflict".
In reality he has been publicly humiliated by Mr Yanukovich's offer at the weekend to give his job to former economy minister Arseny Yatsenyuk, another opposition leader, in an effort to stem the rising protests against his rule.
The opposition has been calling consistently for the resignation of the Azarov government since the onset of the crisis.
But opposition leaders have shied away from the offer of top government posts by Mr Yanukovich, regarding it as a trap intended to compromise them in front of their supporters on the streets.
Mr Yatsenyuk, one of a "troika" of opposition leaders, formally turned down the offer of the top government job on Monday night, and the question now is whether the president would accept Mr Azarov's departure or not.
The steward of the heavily indebted economy through hard times and recession, Mr Azarov backed the decision in November to walk away from a free-trade agreement with the EU - the move which sparked the mass street protests.
During a stormy parliamentary debate with the opposition it was Mr Azarov who received much of the chamber's anger as he defended the need for closer economic ties with Russia.
Parliament met in an emergency session yesterday with ministers loyal to Mr Yanukovich saying they would press for a state of emergency to be declared if opposition leaders did not rein in protesters and end their occupation of municipal and government buildings across the country.
Yanukovich loyalists - clearly under pressure from the president and his aides to make a U-turn - then voted to repeal the anti-protest legislation they had introduced on January 16.
It was these laws - banning most forms of public protest - which sparked a violent turn on the streets, prompting barricades to be erected across Kiev and leading to clashes between activists and police in which six people were killed.
Opposition leaders sought to keep up the pressure on Mr Yanukovich, with Mr Yatsenyuk calling on him to swiftly sign the repeal of the laws into force.