General Aminullah Patiani, the senior Afghan commander in the operation, echoed that "all of the areas of Marjah and Nad Ali have been taken by combined forces".
James Bays, Al Jazeera's correspondent in Kabul, said: "Much of Marjah is safe enough for dignitaries to visit. Forces have set up a cordon of three kilometres around the town and they are trying to extend that cordon."
But the offensive, known as Operation Moshtarak, was overshadowed on Sunday by the death of 12 Afghan civilians killed when two rockets missed their target and landed on homes in Nad Ali district, where Marjah is located. Nato acknowledged responsibility for the deaths.
General Stanley McChrystal, the head of US and Nato forces in Afghanistan, called the loss of life "regrettable" and said the operation was being conducted with "the protection of Afghan people in mind".
"We extend our heartfelt sympathies and will ensure we do all we can to avoid future incidents," he said in a statement on Sunday.
Richard Holbrooke, the US special envoy to Afghanistan, told Al Jazeera that "General McChrystal has gone out of his way to minimise civilian deaths".
Speaking to Al Jazeera while on a trip to Qatar, Holbrooke said: "The objective here is to protect the people and help the government provide services to the people of Afghanistan, even in the most difficult areas."
Operation Moshtarak, is the first major test of the strategy of Barack Obama, the US president, to reverse the Taliban insurgency and end the eight-year conflict with one of the biggest offensives since the 2001 US-led invasion.
US marines were leading a force of 15,000 US, Nato and Afghan troops in the ground and air operation designed to clear the Taliban from the Marjah region of the southern province of Helmand and make way for Western-backed authorities.
Two Nato soldiers have been killed during Operation Moshtarak and another five Nato soldiers died elsewhere in southern Afghanistan since the assault began on Saturday.
Separately, Mohamed Hanif Atmar, Afghanistan's interior minister, told a press conference on Monday that the Taliban should end its fight and accept the government's proposals for reintegration.
"Today our message to them [the Taliban] is that their best option is to take advantage of the peace and reconciliation programme. There is no way you can win, the Afghan people are determined to win," he said.
"If they choose to take advantage of this programme, we will definitely respond positively."
Afghan officials say they have a "government-in-a-box" ready to sweep in and set up institutional services and security that will ensure the Taliban do not return to areas captured by US-led forces.
Obama has ordered the deployment of over 50,000 American troops to Afghanistan since taking office in January 2009, with the final reinforcements due to bring to 150,000 the total number of US and Nato-led troops in the country by August.