"Boris Johnson says the UK's relations with Russia are "not on a good footing" but he wants them to improve, after talks in Moscow.
But he said he trusted Mr Johnson and they had agreed on the need to work together on the UN Security Council.
Mr Johnson is the first UK foreign secretary to visit Russia in five years.
Mr Lavrov said it was no secret that Britain's relations with Russia were at a "low point".
And he accused Britain of making a series of "aggressive and insulting" public statements ahead of their meeting, saying Russia had done nothing to justify being seen as an aggressor in relation to its actions in Ukraine and Syria.
"I cannot recall any of Russia's actions that would be aggressive in relation to the United Kingdom. We did not blame London for anything," said Mr Lavrov.
"On the contrary, we have heard accusations, even insultingly formulated - that we support the criminal regime in Syria, that we are aggressors, that we are occupiers, we annex other territories.
"And all this despite the fact that on all the regional issues in question, and on many others, all information about what our position is, what it is based on, is regularly provided."
The pair also clashed over Russia's alleged attempts to interfere in elections in the West, following UK Prime Minister Theresa May's warnings about the risks of Russia's "sustained campaign of cyber-espionage and disruption".
Ahead of the meeting in Moscow, the UK government said Mr Johnson would warn Russia to stop cyber-attacks which threaten Britain's national security or face retaliation of a similar kind from the UK.
But Mr Lavrov accused Mr Johnson of being a "hostage" of untrue Western narratives on the issue, insisting Russia had not meddled in elections in other countries.
Mr Johnson said there was "abundant evidence" of Russian interference in polls in the US, Germany, Denmark and France."...
[Ed. note: This claim isn't supported by reported evidence: "German Intel Clears Russia on Interference," Consortium News, R. McGovern, 2/15/2017; "German Election Mystery: Why No Russian Meddling?" NY Times, 9/21/2017; "The Latest: France says no trace of Russian hacking Macron," AP, 6/17/2017; In Denmark, no election "hacking" reported, despite headline. Attempted and failed email spear phishing isn't "hacking." Spear phishing email attempts to Defence Ministry in 2015 and 2016 were largely unsuccessful, none successful vs classified information: "October 2016: The hacker’s third attempt at stealing login information using a fake login page is attempted and about 1,000 phishing emails were sent to specific employees working for the Defence Ministry." For perspective, the US state of Wisconsin receives 60,000 "attempts" against its computer systems daily: "Two days before the state's April 5  presidential primary last year, security analysts logged more than 150,000 alerts in a single day. On a typical day over the past two years, the state has logged fewer than 60,000 of these alerts, which attempt to find holes in state systems." 9/26/2017, "In reversal, feds proclaim Russians did not seek to hack Wisconsin's election system," jsonline....No computer is guaranteed secure. Don't use computers for non-public material. As to the US, it has long history of meddling in foreign elections: 10/13/2016, "The long history of the U.S. interfering with elections elsewhere," Washington Post,
(continuing): "Mr Lavrov hit back by telling Mr Johnson he himself had said Russia had not interfered in Britain's general election and Brexit referendum.
Mr Johnson interrupted his Russian counterpart to add: "Not successfully."
Mr Lavrov said the evidence produced so far of Russian attempts at interference amounted to no more than the spending of "a few kopecks" on social media adverts.
'Cradle of democracy'
"I think you have made all this up in your Western community and unfortunately right now you are hostage to this subject, it is very difficult for you to climb down from the fence you have climbed."
He also criticised Britain for cutting off ties with Russia's FSB security agency over the murder of Alexander Litvinenko in London, saying the UK authorities had refused to hand over information in the case.
He said government criticism of British politicians who speak to Russian media outlets, such as the RT television channel, damaged the reputation of the UK as "the cradle of democracy".
Mr Johnson acknowledged the "difficulties" in relations with Russia, adding: "It is a regrettable state of affairs but it should not preclude co-operation."
The UK foreign secretary said they had identified common ground on issues such as North Korea, Syria and trade - and said the UK and Russian security services should co-ordinate ahead of next year's World Cup.
As the mood at the press conference relaxed, Mr Lavrov said: "I trust Boris and I trust him to an extent that I am ready to call him BorIs [Russian-style pronunciation] rather than BOris."
Mr Johnson said he adopted the approach Ronald Reagan had taken with Mikhail Gorbachev: "Trust, but verify."
And he joked that his trust was so great that he had handed his coat with "everything in my pockets, secret or otherwise" to Mr Lavrov when he arrived at the Ministry of Foreign Affairs building.
Mr Lavrov joked back: "I can say that there was nothing in the pockets of Boris's coat", to which Mr Johnson responded in surprise: "So you have searched it already?"
Mr Johnson's trip follows Prime Minister Theresa May's accusation last month that Russia was trying to "undermine free societies".
Her criticisms were repeated by Ciaran Martin, chief executive of GCHQ's National Cyber Security Centre, who said that Russia was "seeking to undermine the international system"."
[Comment: What is "the international system?"]
"Analysis by BBC diplomatic correspondent James Landale:"
"Despite the differences between London and Moscow, both sides have an interest in improving what is a poor relationship.
There are several issues where both Britain and Russia sometimes disagree but want more dialogue.
On Syria, the UK wants to help shape any future political settlement while Russia needs western money to help rebuild the country.
On North Korea, both Russia and the UK want to find ways of de-escalating the crisis prompted by Pyongyang's ballistic missile programme.
And on Iran, both sides want to do what they can to protect the deal they helped negotiate to curb Tehran's nuclear ambitions.
So Friday's meeting may have allowed both sides to rehearse their differences - and the veteran Russian foreign minister, Sergei Lavrov, gave as good as he got from the comparative novice foreign secretary, Boris Johnson.
But it also allowed them to crack a few jokes and build a relationship that they could need in the years to come.
This was not a reset or a return to business as usual but the opening of a channel of communication that in recent years has been as frozen as the Moscow winter."