His comments came weeks after Theresa May accused President Vladimir Putin of using cyber-espionage and disruption to sow discord in the West, warning Moscow: "We know what you are doing. And you will not succeed."
And they follow GCHQ's revelation on Wednesday that Britain has developed sophisticated cyber weapons capable of crippling a hostile state.
Theresa May: "I have a simple message for Russia: We know what you are doing. And you will not succeed."
Parliament's Intelligence and Security Committee was told the agency was developing a "full spectrum" of cyber weaponry.
Mr Johnson is the first UK Foreign Secretary to visit Russia in five years, after a period in which Moscow's relations with the West have been plunged into the deep freeze by the illegal annexation of Crimea and alleged use of social media to influence democratic elections.
Speaking to Mr Lavrov in Moscow, the Foreign Secretary will make plain that his visit does not represent an attempt to return relations to "business as usual".
But he will also say he wants the kind of mature dialogue between the two permanent members of the UN Security Council that will allow them to work together on issues of mutual interest such as preserving the Iranian nuclear deal, pushing forward a political settlement in Syria and countering the threat to global stability from North Korea.
Speaking on the eve of the meeting, Mr Johnson left no doubt Britain had made preparations to respond to continued misuse of cyberspace.
"Over the last year or 18 months we have fully taken on that argument," he told reporters.
"The UK is certainly prepared and able to respond should we so desire.
"I can assure you that the UK is a world leader in this field. We don't conduct malign or disruptive cyber activity, but the logic of deterrence is clearly something we appreciate in the cyber field.
"That is something we have acquired and are acquiring."
Mr Johnson said relations between the UK and Russia "haven't been so bad for a very long time".
"There are areas in which Russia is behaving in a more hostile way towards our interests than at any time since the end of the Cold War," he said.
"I will make clear to Lavrov that there are things that we find extremely difficult to accept, and we can't accept."
The occupation of Crimea, continuing destabilisation of eastern Ukraine and the western Balkans and attempts to influence democratic processes in the West are all "obstructions in our relationship that can't be ignored", he said.
But Mr Johnson will say that these disagreements should not prevent the two countries cooperating on global issues of shared concern.
In face-to-face talks with Mr Lavrov, he will evoke the Second World War alliance between Britain and the Soviet Union as an example of working together despite profound differences.
The Foreign Secretary said: "Our relations with Russia cannot be 'business as usual' whilst Russia continues to attempt to destabilise European states, including Ukraine.
"However, it is vital for international security that we do talk to each other, as the consequences of miscommunication or misunderstanding are grave.
"My visit to Russia comes at a critical time as we need to work together to solve the world's most pressing global challenges."
The UK was a firm supporter of the European Union's decision last week to roll over sanctions on Russia and continues to insist that Moscow must live up to the terms of the Minsk Agreement requiring it to cease support for armed separatists.
Mr Johnson will indicate to Mr Lavrov the UK would be responsive to a change in stance.
"We have a relationship with Russia that spans over 450 years," he said. "Our similarities and historical links far outweigh our current political disagreements.
"The Kremlin has positioned Russia in direct opposition to the West, but it doesn't have to be that way."
The two men will discuss ongoing preparations to ensure the safety of thousands of England football fans expected to visit Russia for next year's World Cup, which have already seen reciprocal visits by police from both sides.
Amnesty International UK's head of policy and government affairs, Allan Hogarth, urged Mr Johnson to use the trip to put on record his opposition to Russia's notorious "gay propaganda" law.
"We're only months away from the World Cup, and LGBTI fans - from Britain and around the world - will be travelling to a country where homophobia is officially legitimised, and verbal and physical attacks on people because of their perceived sexual orientation are commonplace," said Mr Hogarth.
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