Nato Secretary General Anders Fogh Rasmussen says the decision is a "significant step"
Nato's Secretary General, Anders Fogh Rasmussen, has said it has decided to take on the whole military operation in Libya "with immediate effect".
The alliance will enforce "all aspects" of the UN resolution authorising action to protect civilians. "Nothing more, nothing less," Mr Rasmussen added.
Libyan rebels have been advancing westwards, capturing towns abandoned by Col Muammar Gaddafi's forces.
Explosions were also heard in Sirte and the capital, Tripoli, on Sunday night.
It is not clear what the causes of the blasts were, though state TV said the cities were being bombed by "Crusader and colonialist" forces. A government spokesman also said the town of Sabha had been targeted.
Sirte, the Libyan leader's stronghold, is only 100km (60 miles) west of the town of Nufaila, which rebel forces said they had reached. Foreign journalists said the city was swarming with soldiers on patrol.
The rebels earlier retook the eastern coastal towns of Ras Lanuf, Brega, Uqayla and Bin Jawad, only a day after seizing control of Ajdabiya.Disagreements
Nato's plan to take responsibility for operations in Libya had already been agreed by military representatives of the 28 member states, but it needed ambassadors to provide political approval at a meeting in Brussels.
In a communique hailing the "very significant step", Mr Rasmussen said that in the past week the alliance had "put together a complete package of operations in support of the United Nations resolution by sea and by air".
"We are already enforcing the arms embargo and the no-fly zone, and with today's decision we are going beyond. We will be acting in close co-ordination with our international and regional partners to protect the people of Libya."
Mr Rasmussen said Nato's goal was to "protect civilians and civilian-populated areas under threat of attack from the Gaddafi regime".
"Nato will implement all aspects of the UN Resolution. Nothing more, nothing less," he added.
End Quote Robert Gates US Defence Secretary
[Col Gaddafi's] ability to move armour, to move toward Benghazi or a place like that, has pretty well been eliminated”
Nato's top operational commander, Gen Charles Bouchard of Canada, would "begin executing this operation with immediate effect", he concluded.
The BBC's Chris Morris in Brussels says the mission to protect civilians was more sensitive because it involved debate about what exactly are legitimate military targets on the ground.
There were disagreements notably between France and Turkey about political control of the mission, but they have now been resolved, our correspondent says.
But the precise rules of engagement have not been revealed, he adds.
Alongside the Nato command structure will be a separate, high-level committee of representatives of all countries taking part in the military action, including Arab states. It will give what one official called "broad political guidance."'Under pressure'
While Nato ambassadors discussed the international military operation, rebel forces in eastern Libya took advantage of the devastating effect of the air and missile strikes on Col Gaddafi's forces to advance westwards.
"Gaddafi's forces are now scared rats," Mohammed Ali al-Atwish, a rebel fighter in Bin Jawad, told the AFP news agency.
At the scene
In the last 24 hours, the rebels have pushed hundreds of kilometres to the west. The next big city in their path is Sirte. It is Col Gaddafi's hometown and one of the very few places untouched by the spirit of rebellion.
So here is the dilemma: if the rebels do manage to get that far and the people of Sirte do not rise up, either because they are loyal to Gaddafi or too afraid to act, what do the allies do? If civilians are not being threatened, they arguably have no mandate for action and that would stall the rebels advance and leave them exposed and vulnerable to attack and that could mean an open-ended engagement for the coalition.
If the coalition launches attacks anyway to weaken Col Gaddafi's forces, that will convince many that this really is about regime-change and that could create splits within the alliance. The rebel advance may be quietly cheered in London, Paris and Washington, but it also potentially brings a host of problems for the coalition.
"They are dropping their weapons and uniforms and dressing as civilians. We are no longer concerned about Gaddafi's forces at all."
The BBC's Ben Brown in Ras Lanuf says the rebels are in a state of high excitement, and can hardly believe the progress they have made.
They claim that they could be in Sirte by Monday, but the further they advance towards Tripoli, the greater the fight the regime is likely to put up, our correspondent says.
In interviews with US media on Sunday morning, US Defence Secretary Robert Gates that Col Gaddafi's "ability to move armour, to move toward Benghazi or a place like that, has pretty well been eliminated".
"Now we'll have to keep our eye on it, because he still has ground forces at his beck and call. But the reality is that they are under a lot of pressure."
Mr Gates also said there was a political push to ease the Libyan leader from power, and that it was possible that more of his associates would defect.
"We have things in our tool box in addition to hammers... one should not underestimate the possibility of the regime itself cracking."
Libyan officials say the strikes have killed nearly 100 civilians but this cannot be independently confirmed.
Late on Sunday, government spokesman Moussa Ibrahim said three young Libyan men had been killed in an air strike on a fishing harbour near Sirte. There was "nothing military or quasi-military" there, he said.
Mr Ibrahim also told reporters that a "peace convoy", which had been heading from Sirte towards Ajdabiya and Benghazi, was attacked by rebels near Bin Jawad. About 29 people were injured in the "very cowardly act, witnessed by thousands of people," he said.
Finally, he said a woman who stormed into a hotel in Tripoli on Saturday to tell journalists that government troops had raped her, Iman al-Obaidi, was now with her family. Four men, including the son of a high-ranking police officer, had been questioned about her allegations, he added.
Meanwhile, Col Gaddafi's troops have continued heir bombardment of Misrata, the only significant rebel-held city left in the west.
On Sunday evening, a resident told the BBC that eight people had been killed and 26 wounded - five of them critically - as Col Gaddafi's forces advanced on the al-Jazeera residential area in the west of the city.
"They used mortars and heavy anti-aircraft guns," he said. "The injuries are mainly from the explosions, I am talking about severed limbs and big injuries in the trunk area. There are also crush injuries due to the collapse of buildings."
A rebel spokesman told the BBC that fighting was continuing for control of the main road through the city.
Libyan state TV earlier said Misrata was "secure" and life was "going back to normal". Security forces had arrested "terrorist gangs", it said.
Meanwhile, the Benghazi-based Transitional National Council said the rebels could begin exporting oil in less than a week.
Spokesman Ali Tarhouni said oil fields in territory under opposition control were already producing more than 100,000 barrels of crude a day.
He said the Gulf state of Qatar had agreed to help bring it to market.