Two embassy staff members and at least four Afghan civilians died in the blast, a Russian state news agency and a local police official said.
The death toll is expected to rise, with some reports saying up to 10 people had been killed.
The RIA Novosti news agency said the explosion went off when a Russian diplomat came out to call out the names of candidates for visas to people waiting outside.
The suicide bomber was shot dead by armed guards as he approached the embassy’s gate, claimed officials. It was not immediately clear if he was able to detonate the explosives.
Students seeking Russian visas were in the area during the attack.
Mawlawi Sabir, the head of the police district where the attack took place, told Reuters: “The suicide attacker before reaching the target, was recognised and shot by Russian embassy (Taliban) guards ... there is no information about casualties yet.”
Officials said an investigation had been launched into the tragedy.
Russia’s Investigative Committee confirmed that the two killed Russian embassy staff members were the embassy’s second secretary and a security guard.
In a statement, the Investigative Committee, the state body that probes major crimes, said that its chairman Alexander Bastrykin had ordered an investigation to be opened.
The four others killed were Afghan civilians, Khalid Zadran, a Kabul police spokesman said.
Russia is one of the few countries to have maintained an embassy in Kabul after a Taliban resurgence saw them take over the country in August 2021 as Western allies withdrew from Afghanistan.
Although Moscow does not officially recognise the Taliban government, they have been in talks over an agreement to supply commodities including gas.
According to a statement by the Russian foreign ministry, the explosion occurred “in the immediate vicinity of the entrance to the consular section of the Russian Embassy” in the Afghan capital.
“An unknown militant set off an explosive device,” the ministry said. “As a result of the attack, two members of the diplomatic mission were killed, and there are also victims among Afghan citizens.”