The Afghan Government has described the meeting as "not in the interest of Afghanistan and the efforts for peace process", and it accused Moscow of failing to fulfill its past pledge to facilitate direct talks between its officials and Taliban members.
"The interim government should also pave the way for a transparent election", Atta Muhammad Noor, an influential former warlord, said on Twitter.
Abdullah Abdullah, the country's chief executive, said Monday that the Afghan government should be at the centre of any peace talks, adding that Kabul "would prefer the Moscow meeting had a different shape".
Such a meeting between the Taliban and Afghan politicians - including Karzai, who was appointed by the USA - is nearly unheard of.
The Russian Embassy in Kabul issued a statement late on Saturday on behalf of the "Afghan Society of Russia".
Participants attend the opening of two-day talks between Taliban and Afghan opposition representatives, at the President Hotel in Moscow, Russia, Feb. 5, 2019. While Khalilzad has stressed that much remains to discuss, the broad goal would be for the United States to pull troops from Afghanistan and for the Taliban to ensure that foreign extremists will not be based on its territory - the reason for the USA invasion after the September 11, 2001 attacks.
"But after ending their military presence, their non-military teams can come and we need them too, they can come and take part in the reconstruction and development process", he said.
The group was notorious for its treatment of women, banning a lot of them to work or go to school.
"We all of us agreed on the withdrawal of foreign forces and the establishment of an Islamic government and making a constitution - an Islamic constitution", he said. President Ashraf Ghani insists that a final agreement must protect the inalienable rights and values of Afghans.
The Taliban's longtime negotiator, Mohammad Abbas Stanekzai, who has taken part in all rounds of discussions between the insurgents and Khalilzad among other U.S. diplomats in recent months, will lead the Taliban delegation in Moscow.
Those efforts culminated in six days of talks between the United States and the Taliban in January where both sides touted "progress" - stoking Afghan fears that America could cut a deal to withdraw its troops before a lasting peace with Kabul is reached.
Afghan Foreign Ministry Spokesman Sebghatullah Ahmadi told VOA the Moscow meeting has no legitimacy and the government will not accept any possible outcome of the discussions.
The Moscow talks are officially being organised by members of the Afghan diaspora.
"The people of Afghanistan want a country with sustainable peace, a united and democratic Afghanistan, a prosperous Afghanistan built by the educated women and men of it", he said.
Khalilzad is in a hurry to find a peace deal for Afghanistan that would allow America to bring home its troops after 17 years of war.
However, Mr Stanikzai said "women should not worry" about the prospect of increasing Taliban influence as they would seek to grant women all their "rights according to Islamic rule and Afghan culture".
But images of Karzai and other powerful leaders attending prayer lead by a Taliban figure and dining with the militants invoked anger in Afghanistan.
The militants have indicated they would provide a safe environment for women's work and education under an "Islamic system" they have proposed for Afghanistan's future.
Taliban officials have used the Moscow meeting to outline their vision for a post-war Afghanistan.
"Moscow does not believe in Ghani's future", said Serenko of the Centre for Contemporary Afghan Studies, suggesting Russian Federation was backing Atmar in the run-up to the polls.
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