From BBC Football:
The Sky Blues had until 1600 GMT on Friday to complete a deal and prove their solvency and they left it late.
"The successful conclusion of this transaction ensures our future is secure and the threat of administration is avoided," said a Coventry statement.
The takeover has allowed Ranson to become the Sky Blues' new chairman.
Ranson, who played for Manchester City and Birmingham City, has been linked with takeovers for Man City, Aston Villa and Southampton in the past.
Major shareholders Craigavon and Sir Derek Higgs - who own 71.4% of the club's shares - have agreed to the offer and the takeover is subject to the acceptance of 90% of all shareholders.
The offer will be forwarded to the remaining 28.6% of shareholders later on Friday.
Retiring chairman Joe Elliott admitted he was delighted to welcome hedge fund Sisu Capital and Ranson to the club.
"This is a very exciting day in the long, proud history of Coventry City Football Club and I am positive that will be reflected in the attendance for Saturday's game against Southampton," said Elliott.
"I am sure that Ray Ranson and Sisu Capital will help drive Coventry City forward into a brand new era for the club.
"I'd like to pay special thanks to Iain Dowie, his management team, the players and all of the staff at Coventry City who have loyally supported the club through such a difficult time."
Dowie - who had suggested he might leave Coventry should they go into administration - revealed his relief that the club's future was now more secure.
"This takeover can only benefit Coventry City Football Club and I look forward to sitting down with Ray and discussing the way forward," said Dowie.
"These are exciting times for everyone connected with the Sky Blues and I am delighted that the threat of administration - and all of the uncertainty surrounding it - has been lifted."
Sisu - who made an unsuccessful bid for Southampton last month - have told supporters that Coventry's reported £38m debt will be restructured to ensure ongoing financial and operational commitments are met.
Resources up to the value of £20m have also been promised, with Ranson named as Coventry's new chairman.
The existing Sky Blues board will resign from their positions ahead of new appointments being made.
"Both Sisu and Ray Ranson would like to take this opportunity to thank the existing board for the services they have provided to the club and their valued support during this acquisition process," read a statement.
Sisu have also signed a co-operation agreement with stadium owners ACL for the continued use of the Ricoh Arena.
Ranson's Sisu Capital group looked to have competition for the club from Greek billionaire Alki David, but the filmmaker withdrew his interest earlier this week.
David, with a reported fortune of £4bn, had claimed that the club was "falling apart".
From BBC Football:
But the Sky Blues, who have debts of £37m, remain hopeful that a takeover by entrepreneur Ray Ranson and Sisu Capital will be completed before then.
"This legal notice gives Coventry a 10-day window to complete the takeover talks," said the board's statement.
If the talks fail and Coventry do go into administration, they will be deducted 10 points.
But chairman Joe Elliot moved to calm fans' concerns about Monday's annnouncement.
"We've done this because the clock has been ticking for a long time," he told BBC Radio 5 Live.
"We're in very serious negotiations with Ray Ranson and Sisu Capital and we're convinced we can bring it together quickly.
"There is now a 10-day deadline on it because we've filed notice of intent but that doesn't mean we're in administration, it means if things don't go right, we will be.
"We've got to push forward as quickly as possible to get ourselves a deal and take Coventry back into the Premier League."
Coventry lie 14th in the Championship and their perilous financial state led to a transfer embargo being imposed.
Monday's statement said: "This is merely a legal process protecting the bank and its creditors.
"It should speed up the takeover of the club and therefore take Coventry into a solvent financial situation.
"Following recent speculation, the club can confirm can that staff and player wages have now been paid and we look forward to a successful conclusion of the takeover in the near future."
Manager Iain Dowie has warned that he would consider his position if the club went into administration but he remains positive.
"The sword of Damocles has been hanging for a little while now," he told 5 Live.
"When I came to the club, I thought it would be a very positive situation where we'd have quite a sizeable amount of money to spend. It hasn't quite turned out like that.
"We'd like to be able to take this club forward and we're a little bit frustrated that some of our targets are now playing very well elsewhere.
"But we can throw our toys out of the pram or work as hard as we can, and I'll choose the latter."
Elliot said Coventry's financial situation was exacerbated by their relegation from the Premier League.
"We came down from the Premier League with £60m of debt," he explained.
"We reduced it but it's still very difficult to cope when you're running with a debt of £36m or £37m - especially as we don't own the stadium.
"But hopefully this is the start of 10 jolly good days. I'm really hopeful - I've just spoken to Ray Ranson and he's very hopeful - we just need to get things moving along.
"The stakes are high and we're working as many hours as it takes to get this deal done."
Coventry's Ricoh Arena stadium was jointly funded by the council and the Alan Higgs Trust - a charity run by businessman Derek Higgs - and is run by Arena Coventry Limited (ACL).
The club have not filled the stadium's capacity of 32,600 since moving there in August 2005.
IT company Ricoh earned naming rights to the stadium with a sponsorship deal worth a reported £10m.
From Coventry Observer, by Nick Connoll:
The Sky Blues faithful were hopeful of a decision over the club's future after the weekend.
But any hopes were dashed when club chairman Joe Elliott said the meeting was scrapped in favour of a series of shorter get-togethers between board members and other involved parties.
Daniel Allvey, chairman of Arena Coventry Limited which own the Ricoh Arena, called on the club to make a decision last week amid fears the prolonged wait could force the club into administration.
The club are currently discussing potential bids by former footballer and wealthy businessman Ray Ranson and Coventry MP and former Sky Blues chairman Geoffrey Robinson.
The urgency of Mr Allvey's statement suggested the club's board could have been pressured into a decision over the weekend.
But Mr Elliott's announcement there was no meeting at all turned out to be the exact opposite of what fans were hoping for.
The extremely complex nature of any takeover deal for the club and the Ricoh Arena has left a number of obstacles in the way of any potential investors.
The club's reported £38 million of debt would have to be taken on by buyers as well as providing sufficient funds to help manager Iain Dowie strengthen the squad that has so far managed to reach eighth in the Coca-Cola Championship.
But the main sticking point from months of negotiations, stretching back to last season, has been the acquisition of the Ricoh Arena from Coventry city council.
With the club losing around £90,000 a week, the stadium is integral to any deal.
But so far the council have had to seek a deal that would be beneficial to the city while also seeking assurances from any potential buyers over the regeneration of the north of the city.
See also: Elliott talks down administration threat
