The attack was claimed by Islamic State militant group on its channel on Telegram.
Police said the attacker was shot dead by armed guards as he approached the gate, in one of the first such attacks since the Taliban took power last year.
"The suicide attacker before reaching the target, was recognised and shot by Russian embassy (Taliban) guards ... there is no information about casualties yet," Mawlawi Sabir, the head of the police district where the attack took place, told Reuters.
The Russian Foreign Ministry said in a statement that an unknown militant set off an explosive device near the entrance to the consular section of the embassy around at 10.50 am Kabul time.
"As a result of the attack, two employees of the diplomatic mission were killed, and there are also victims among Afghan citizens," the ministry said.
The four others killed were Afghan civilians, Khalid Zadran, a Kabul police spokesperson said.
Russia is one of the few countries to have maintained an embassy in Kabul after the Taliban took over the country more than a year ago. Although Moscow does not officially recognise the Taliban's government, they have been in talks with officials over an agreement to supply gasoline and other commodities.
The United Nations' mission in Afghanistan (UNAMA) condemned the blast.
"In light of recent events, UNAMA stresses the need for the de facto authorities to take steps to ensure the safety and security of the people as well as diplomatic missions," the UN wrote on Twitter, in reference to the Taliban government.
The spokesperson for the foreign ministry in Afghanistan said Taliban forces would take serious steps towards securing embassies operating in the country.
"(The government) has a close relationship with Russia; we will never let such negative actions of enemies have a negative impact on the relationship," spokesperson Abdul Qahar Balkhi said in a statement.
During the decades-long Taliban insurgency against the Western-backed Afghan government, bombings targeting foreign missions were a regular occurrence in Kabul, especially in recent years, with embassies and hotels fortifying themselves with razor wire and blast walls.
Such incidents have decreased dramatically since the insurgent group swept to power in August 2021.
However, the Taliban has had to contend with an insurgency by the Afghanistan affiliate of the Islamic State militant group.
The attacks had thus far been against the Taliban and civilian targets such as mosques.
