Opener Cook had come under increasing pressure in recent months due to his own poor form and a string of disappointing results in the 50-over game.
A 5-2 series defeat against Sri Lanka, in which Cook made just 119 runs from six innings with a top score of 34, represented the tipping point for the selectors and the England and Wales Cricket Board (ECB) released a statement on Saturday to formalise news that had been widely reported the previous day.
Cook will now miss out on the Cricket World Cup that begins in Australia and New Zealand in February, with Morgan - whose own form in 2014 has been similarly unimpressive - taking the reins ahead of next month's tri-series with Australia and India.
In a statement on the ECB's official website, national selector James Whitaker said: "We spent a considerable time yesterday discussing the make up of a squad that gives England the best possible chance of success at the World Cup which begins in less than two months.
"Having reviewed the recent series against Sri Lanka we came to the conclusion that there was no place for Alastair Cook amongst our strongest 15 one-day players and therefore recommended Eoin Morgan be appointed one-day captain."
Paul Downton, the managing director of England Cricket, added: "I have complete confidence in the decision made by the selectors and having spent time with Alastair yesterday he accepts that it has been made in the best interests of the one-day side."
Cook, who will continue to lead England's Test side, was handed the ODI reins in May 2011 and led England to the summit of the world rankings the following year.
However, the fortunes of Cook and the team have declined sharply since they suffered defeat to India in the final of the 2013 Champions Trophy.
England have not won an ODI series since, while Cook has made just one half-century in 22 subsequent innings and faced criticism for his scoring rate and captaincy.
The departing ODI skipper said: "I am gutted to be left out of the World Cup squad and it is likely to take me a while to get over the disappointment. That said I wish [head coach] Peter Moores, Eoin and the lads all the best for the World Cup.
"I would also like to thank all the players and fans who have supported me during my time as one-day captain."
Morgan has played 107 ODIs for England, including eight as captain, after initially representing Ireland - the country of his birth - at international level.
He said: "It is a huge honour to captain England and I am delighted to be leading the one-day side. I firmly believe that with the players currently involved in the one-day set up we have the makings of a very good one-day side, a young side that can surprise people at the World Cup.
"We all recognise we still have a lot of work to do on our one-day game but the potential of this group of players is huge and it is now up to us as players to turn that potential into consistent performances on the international stage starting next month against Australia and India."
England will announce their squad for the World Cup and tri-series against Australia and India later on Saturday.