It is the largest combat operation since Barack Obama, the US president, ordered 30,000 US reinforcements to Afghanistan last December.
Danish, Estonian and Canadian troops are also involved in the campaign.
Soon after the offensive began, five Taliban fighters were reported killed.
And by the first day's end, one service member of the International Security Assistance Force (Isaf) had died in a bomb attack while another was killed by gunfire, according to a spokesman for the Nato-led multinational force.
Nato declined to give their nationalities.
Separately, the UK defence ministry announced the death of one British soldier from the 1st Battalion Grenadier Guards in an explosion while on vehicle patrol in Nad-e-Ali district of Helmand.
Elsewhere in southern Afghanistan, a suicide bomber detonated an explosive device on a motorbike near US and Afghan troops on a joint foot patrol in Kandahar province on Saturday.
A police commander told Al Jazeera that two children were killed and that US forces also suffered casualties in the attack, which took place in Arghandab district, northwest of Kandahar city.
Fight for Marjah
Marine commanders say they expect anywhere between 400 to 1,000 fighters to be holed up inside Marjah, a town of 80,000 people, including more than 100 foreign fighters.
But Taliban sources have insisted the number is closer to 2,000.
Qari Yousef, a Taliban spokesman in the south, told Al Jazeera that foreign forces had been bombarding the area around Marjah for days and that the operation had in fact begun on February 7.
He also warned that the Taliban would offer stiff resistance.
"Our decision is that there will definitely be resistance because foreign invaders have come to invade our country," he said.
"If they need 15,000 troops to take over a small village - what will they need to take over a province which is under the Taliban's hands," he told Al Jazeera.
Al Jazeera's Zeina Khodr reporting from Kabul, the capital, said: "We've been talking to the Taliban over recent days and they are making clear that they will defend the territory and fight till death."
"Foreign troops want to also send a clear message that the Afghan government will re-establish government presence in Marjah and separate the town from the Taliban to improve people's lives, open roads and government institutions, which is all part of the new Obama strategy being employed in the region," she said.
"It's not all military tactics because Marjah is really strategic, it's at the doorstep of Lashkar-Gah, the provincial capital of Helmand and if you open those roads you can improve economic development for the people, but they are worried, mostly about civilian casualties."
Isaf termed the offensive a "clearing" operation to be followed by "smaller-scaled 'shaping' operations".
Mohammad Gulab Mangal, the governor of Helmand, said earlier this week that local authorities were poised to move in behind the military operation to set up civil services, including police and security.
|Afghan officials say they hope civilian deaths will be avoided by publicing Moshtarak [AFP]
But Gilles Dorronsoro, a scholar at the Carnegie Endowment for International Peace, said the plan's key weakness was that it failed to include a long-term plan to prevent the Taliban from returning to the area.
"The Afghan state is just a network of warlords [and] opium dealers - to think that these people are going to take Marjah and build a solid state there, I don't think so," he told Al Jazeera.
Marjah is at the heart of Afghanistan's opium production. Fighters there have exploited an irrigation system built in the 1950s with US aid aimed at turning the central Helmand River valley into Afghanistan's bread basket.
Janan Mosazai, an political analyst in Kabul, told Al Jazeera that while a military win in the area seemed assured, it would be the second stage of the operation that would be crucial.
"The test to this new approach ... will come when the operation is over - when the military stage is over - when there is a requirement for Afghan civilian authorities and for reconstruction specialists to move into Marjah and ... give the people of this area the confidence that this time it's different," he said.
"[They must show] that there will be a cleaner, more efficient, less corrupt government put into place and that there will be a police force that is not corrupt and doesn't scavenge on the local population ... [and] that this will be a fundamental, long-term change."
Exodus of civilians
Isaf and the Afghan government have stressed that they hope civilian casualties will be avoided, publicising their operation in advance.
Hundreds of civilians fled the area, but many have stayed.
Jamil Karzai, the head of the Afghan government's commission for national security, and a relative of the Afghan president, said that publicising the operation so heavily in advance had given away military advantage.
"They are just coming to the media and talking to the media and letting their enemies know there is a big operation against them ... everyone out in the country knows about this operation and of course the Taliban and al-Qaeda left the area," he told Al Jazeera.
He said that long-term success would only come from having Afghan forces playing a lead role in any assault.
"Our Afghan forces understand the ground realities, they understand the region, they understand how to fight al-Qaeda and [the] Taliban. When our army or our police are supported from international forces - by American forces - when they are on the ground ... we will win," he said.
"If international forces are ahead, on the frontline, and our national forces are are in the backstage, we will never win this war."