Another defence ministry statement said two Afghan soldiers were "martyred".
Elsewhere in the country, four Nato soldiers died and two were wounded when a roadside bomb exploded.
The British defence ministry confirmed the deaths in Helmand province on Wednesday, saying that one of those who died was a female soldier.
She is the first British female soldier to die in Afghanistan.
The Arghandab operation began at 8am local time [0300 GMT], the Afghan defence ministry said.
Mark Laity, a Nato spokesman in Kabul, told Al Jazeera: "The operation is pretty much on track.
"There have been a number of engagements with the insurgents, but they have been minor rather than major battles.
"We have killed some insurgents but we have not yet suffered any Isaf [the International Security Assistance Force] casualties."
The number of troops involved in the operation is "substantial", Laity said.
"The bulk of the troops are from the Afghan army. They are leading that operation and we are backing them."
Al Jazeera's Hashem Ahelbarra, reporting from Kandahar, said that the main road leading to Arghandab was blocked and that more army reinforcements were heading towards the district, including tanks, armoured vehicles and soldiers.
"The threat from the Taliban in Arghandab is being taken very seriously by the Afghan army and the coalition forces," he said.
"They are concerned that with the military operation going on, the Taliban could retaliate using suicide bombers and attack inside [the city of] Kandahar."
Zemarai Bashari, spokesman for Afghanistan's interior ministry, said "the operation is going very well".
"Afghan security forces have made progress ... the initial reports are stating that 16 enemies have been killed and around four others are injured."
He said that Afghan and Nato troops were carrying out the operation as a joint force with neither leading the other.
"This is a joint operation … everyone is playing their role and we are seeing very good co-ordination and implementation of this operation," Bashari said.
However, Yousuf Ahmadi, a Taliban spokesman, denied that his group's fighters had been dislodged by the Afghan-Nato offensive.
"The fighting started today in the morning but they have not been able to take a metre of the land under our control. We do not intend to leave Arghandab at all," he said.
"We will use Arghandab for specific attacks with motors and cannons on targets in Kandahar city. We have also planned a suicide attack which will be carried out in Kandahar."
Afghan and soldiers from the multinational force soldiers sealed off the Arghandab district on Tuesday, after the Taliban claimed its fighters had taken control of 10 villages.
The Taliban said it met no resistance as it took control of the villages.
Ahelbarra said on Tuesday that hundreds of Taliban fighters had taken up positions in the area.
It was unclear if the group was just trying to make a statement as it did late last year when it captured the same area for just a few days before retreating under heavy bombardment, or if it would try to take back control of large swaths of the country.
|The Taliban freed about 1,000 inmates from |
a prison in Kandahar last week [AFP]
As soon as news of the Taliban takeover circulated, residents fled their villages, some of them with cattle and all their belongings. The Taliban encouraged them to leave."We left the area to protect ourselves from the bombing and the risks of a military confrontation. There are many Taliban fighters - some told us they are more than 800," one resident said.
Gholam Razeq, the district chief of Arghandab, said "the enemy wants to create insecurity in Arghandab which was the most secure area".
Haji Ikramullah Khan, a tribal leader from the region, cautioned that the Taliban fighters could use the cover of the district's grape and pomegranate orchards to mount an attack on Kandahar itself.
"All of Arghandab is made of orchards. The fighters can easily hide and easily fight," he said.
"It is quite close to Kandahar. During the Russian war, the Russians didn't even occupy Arghandab, because when they fought here they suffered big casualties."