US Marine units were twice beaten back by heavy gunfire as they tried to reach a market in the town and another armoured column reported that it had come under fire from three different sniper teams.
"It's a pretty busy day but we expected that because we are penetrating," Lieutenant Colonel Brian Christmas, the commander of 3rd Battalion, 6th Marines, told The Associated Press news agency.
Zeina Khodr, Al Jazeera's correspondent in Kabul, said the resistance, although limited, showed that Taliban fighters were still operating around Marjah.
"Although we are not seeing, for example, face-to-face combat it seems that the Taliban is not going away for now," she said.
"This has been their tactics, ambushing international forces, roadside bombings. The forces are advancing but it is a slow process simply because the area is heavily booby-trapped."
However, Afghan military leaders said that Taliban fighters had fled as about 15,000 US, British and Afghan soldiers had moved in to take control of Marjah and Nad Ali.
"All of the areas of Marjah and Nad Ali have been taken by combined forces. They are under our control, almost all Nad Ali and Marjah," General Aminullah Patiani, the senior Afghan commander in Operation Mushtarak, said.
"The Taliban have left the areas but the threat from IEDs remains."
Rahim Wardak, Afghanistan's defence minister, called on any remaining Taliban fighters to put down their weapons and take up a government reintegration offer.
"I want to call on all Afghan Taliban, the ones who are besieged or maybe hiding ... put down your arms and join our reconciliation programme, take part with us in the rebuilding of our country," he said.
Operation Moshtarak is the first major test of the strategy of Barack Obama, the US president, to take on the Taliban and end the eight-year conflict with one of the biggest offensives since the 2001 US-led invasion.
It is designed to clear Taliban fighters from the Marjah region of the southern province and hold it so that the civilian administration can establish itself.
"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," Richard Holbrooke, the US special envoy to the region, told Al Jazeera.
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.
However, the long-term success of the offensive is likely to depend on securing the support of the local population, something which will be made more difficult by civilian casualties in the offensive.
Nato on Monday expressed its "deep regret" over the deaths of 15 civilians during the offensive in three separate incidents.
Twelve Afghans - six women and six children - were killed when rockets hit their houses suspected to be sheltering Taliban fighters. Three more civilians were shot dead after they ignored warnings from Nato soldiers to stop.
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.