Bosses said all McColl's staff will keep their jobs as the firm's shops transfer to the new owner, while Morrisons will take over the company's two pension schemes.
The convenience chain fell into administration on Friday, plunging the future of its 1,160 shops and 16,000 staff into doubt.
It came after the UK's fourth-largest supermarket chain and forecourt giant EG Group both tabled final offers on Sunday to secure a rescue deal for McColl's.
EG – whose owners also run supermarket giant Asda – had initially been favourites to complete a rescue deal for McColl's.
Morrisons' early approaches had reportedly been rejected by lenders who preferred EG's offer to instantly repay more than £160 million in debts from McColl's.
However, it is understood that Morrisons' successful move will also repay the lenders in cash.
In a statement after the deal was announced, it said that ‘the secured lenders and preferential creditors will be paid in full with a distribution also expected to unsecured creditors’.
Morrisons had also originally proposed to only save the ‘vast majority’ of job and stores, but improved this offer during the bidding process.
The supermarket said: “All McColl's colleagues will be transferred with the McColl's business to Morrisons”.
David Potts, Morrisons chief executive, said: "Although we are disappointed that the business was put into administration, we believe this is a good outcome for McColl's and all its stakeholders.
"This transaction offers stability and continuity for the McColl's business and, in particular, a better outcome for its colleagues and pensioners.
"We all look forward to welcoming many new colleagues into the Morrisons business and to building on the proven strength of the Morrisons Daily format."
A spokesperson for the McColl's pension schemes said: "The trustees welcome the announcement that Morrisons will continue to support the schemes following its acquisition of the McColl's business.
"The trustees will continue to engage with all stakeholders to ensure that members' benefits are protected following the completion of the transaction."
McColl's filed a notice to appoint administrators from PwC last Friday and formally entered administration on Monday.
Morrisons is currently McColl's wholesale supply partner and was expected to immediately terminate its deal with the convenience chain if the takeover move proved unsuccessful
McColl's also runs around 270 stores under the Morrisons Daily brand.
It is understood that administrators were favourable to deal with Morrisons because the Bradford-based supermarket was one of McColl's biggest creditors.
The deal comes less than a year after Morrisons itself was bought for £7 billion by US private equity company Clayton, Dubilier & Rice (CD&R).
McColl's collapse has come after a financial struggle over the past two years as it witnessed soaring costs due to supply chain disruption, inflation and its large debt burden.
Simon Underwood, business recovery partner at accountancy firm Menzies LLP, said: "Morrisons already has a wholesale supply agreement with McColl's, so this may have aided negotiations with the administrators and helped to speed up decision-making on both sides.
"The speed of this deal will come as a great relief to employees, lenders and all creditors with a vested interest in the convenience store chain."