The Russian Football Union (RFU) announced the decision following an executive committee meeting on Tuesday.
It comes after the number of confirmed cases of coronavirus in the country reached 93.
An RFU statement on Tuesday said all matches under their control between March 17 and April 10 would be suspended.
These include the Russian Premier League, the Russian Cup and regional divisions.
League leaders Zenit had been due to face CSKA Moscow away from home on Sunday.
They are nine points clear at the top of the table after 22 rounds of matches.
Farke, who was sacked by Norwich in November after three years at the club, joined Krasnodar in January on a deal that was supposed to keep him there until 2024.
But Russian president Vladimir Putin began a full-scale invasion of neighbouring Ukraine last Thursday, an attack that has been met with international condemnation.
That has been reflected by Russian teams – clubs and national sides – being banned from competing in UEFA and FIFA competitions, including the World Cup.
Markus Gisdol quit his post at Lokomotiv Moscow earlier in the week, confirming to BILD that he felt unable to work in a "country whose leader has invaded another country in the middle of Europe".
He added: "That does not go together with my values and I have therefore resigned as manager of Lokomotiv Moscow with immediate effect.
"I can't stand at the training ground in Moscow and coach the players, ask them to be professional when a few kilometres away there are orders given that brings suffering to the people of a whole country. This is my personal decision and I am absolutely convinced it is the right one."
His compatriot Farke has seemingly taken a similar decision.
While Krasnodar have not outlined the reasoning for Farke and his entire backroom staff leaving, they did confirm the four coaches have left by "mutual agreement".
Farke was due to take charge of his first match at home to Ural next Monday.
The exit of Farke is likely to further increase the pressure on fellow German, Sandro Schwarz, who is Dinamo Moscow coach.
Schwarz, despite seeing his Ukrainian assistant Andriy Voronin leave, is quoted by BILD as saying he plans to stay.
"I'm not the kind of person who buys a ticket, gets on the next plane and takes off. It is not about me. I feel responsible," he said.
CSKA Moscow, Krasnodar, Lokomotiv Moscow, FC Rostov, Rubin Kazan, Spartak Moscow and Zenit are the teams who have been suspended.
On Monday, FIFA responded to Russia's invasion of Ukraine by banning all of the country's national teams and club teams from their competitions. Russian clubs, similarly, have been banned from UEFA tournaments. St Petersburg had already been stripped of this season's Champions League final.
Now the ECA has followed suit, with the seven clubs ceasing to have any involvement with the organisation until further notice.
A statement released by the ECA following a meeting held at Tottenham Hotspur Stadium on Monday read: "Over the past week, ECA has acted swiftly and firmly in both endorsing and participating in the decisions of UEFA to suspend all Russian clubs from participating in UEFA competitions, together with UEFA's decision to end its commercial partnership with Gazprom.
"ECA also strongly endorsed the decision of UEFA's executive committee to move the Champions League final.
"In addition, today, the executive board specifically resolved that all Russian members will cease to be involved in ECA activities with immediate effect until further notice."
The Russian giants confirmed the tragic news on Monday after Samokhvalov suffered suspected heart failure.
Samokhvalov – who had been playing for Lokomotiv's reserve team Kazanka in the third division – was training solo amid the coronavirus pandemic.
"We are truly sad to announce the death of Kazanka defender Innokenty Samokhvalov," Lokomotiv said in a statement.
"The footballer became ill during an individual training session. Details of the tragedy are being clarified.
"Samokhvalov is a pupil of Lokomotiv. Kesha came to our Academy in the third grade, won the youth championship of Russia, this season he played for Kazanka. Samokhvalov left a wife and a son.
"Kesha was a kind, helpful person and a good friend. We are truly shocked by what happened and express our sincere condolences to his family and friends."
