Mr Lavrov (L) said the G8 should not seek to isolate Iran
Foreign ministers from the world's main industrialised countries have issued a statement "deploring" post-election violence in Iran.
The G8's comments were not as strong as France and Italy had wanted, after Russia warned against isolating Iran.
Meanwhile US President Barack Obama praised those who protested against the re-election of hard-line President Mahmoud Ahmadinejad.
He said they had shown "bravery in the face of brutality".
Engagement 'is key'
The G8 foreign ministers, meeting in Italy, said they "fully respect" the sovereignty of Iran, but "deplore post-electoral violence, which led to the loss of lives of Iranian civilians".
"We express our solidarity with those who have suffered repression while peacefully demonstrating and urge Iran to respect fundamental human rights," the statement said.
BBC News, Tehran
The Guardian Council is due to give its definitive verdict on Sunday.
But the remarks by its spokesman are yet another indication that it will be a formality.
The question though is whether the fracture in the ruling elite that this crisis has caused will heal.
When you ask Iranians about the way this might go, a phrase keeps cropping up. They say it might seem quiet to an outsider but there is fire below the ashes.
"The crisis should be settled soon through democratic dialogue and peaceful means."
The G8 called on the Iranian government to guarantee that "the will of the Iranian people is reflected in the electoral process".
They also said the door to dialogue with Iran must remain open.
Russian Foreign Minister Sergei Lavrov said engagement with Iran was "the key word".
He stressed the need to focus on "the main task - to move toward resolving the issues of the Iranian nuclear programme".
He said "no-one" was prepared to condemn the election process, because it was "an exercise in democracy".
The three-day G8 meeting, taking place in Trieste, is the first high-level meeting of the leading Western countries since violence erupted in Iran after the 12 June election.
The primary focus of the G8 talks was intended to be the stabilisation of Afghanistan so Iran, as a neighbouring country, had been invited to attend.
But Iranian Foreign Minister Manouchehr Mottaki withdrew at the last minute.
Before the G8 issued its statement, a spokesman for Iran's top election body, the Guardian Council, said the vote had been fair.
"We have had no fraud in any presidential election and this one was the cleanest election we have had," Abbasali Kadkhodai told the Irna news agency.
The BBC's Jeremy Bowen, in Tehran, says the comments - two days before the Guardian Council's review is completed - suggest their conclusion is a formality.
Meanwhile, a senior hard-line cleric said in his Friday sermon that the leaders of the protests should be dealt with "severely and ruthlessly".
"I want the judiciary to... punish leading rioters firmly and without showing any mercy to teach everyone a lesson," Ahmad Khatami told worshippers at Tehran university in comments broadcast nationwide.
And a war of words between Tehran and Washington appears to be stepping up.
Mr Ahmadinejad had already warned Mr Obama not to interfere in Iran's affairs, after the US leader condemned the Iranian authorities' clampdown on dissent.
But Mr Obama used a joint news conference with German Chancellor Angela Merkel to renew his criticism of the authorities' conduct.
"The rights of the Iranian people to assemble, to speak freely, to have their voices heard - those are universal aspirations," he said.
"Their bravery in the face of brutality is a testament to their enduring pursuit of justice. The violence perpetrated against them is outrageous."
Some 17 people are thought to have died in street protests in the past two weeks, and Tehran has imposed severe restrictions on journalists and the internet.