Abramson's departure was announced yesterday by the US daily's publisher Arthur Sulzberger, and the paper's own report said: "The reasons for the switch were not immediately clear."
Abramson was appointed to head the 160-year-old paper in 2011, and led it in a period during which it was seen as having weathered the transition to digital better than many competitors.
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"Our masthead became half female for the first time and so many great women hold important newsroom positions."
Before taking the top job, the now 60-year-old journalist had been an investigative reporter for the rival Wall Street Journal and then the head of the Times' Washington bureau from 1997.
Her replacement, Baquet, is a 57-year-old newspaper veteran and former editor of The Los Angeles Times.
"It is an honour to be asked to lead the only newsroom in the country that is actually better than it was a generation ago, one that approaches the world with wonder and ambition every day," he said.
Last year The New York Times boasted the largest daily and Sunday circulation of any seven-day newspaper in the United States, with a weekday circulation of 1,926,800 print and online versions.
According to the company's 2013 annual statement, the firm had an annual turnover of $1.57 billion.
But like many dailies, the "gray lady" of US journalism has struggled with the move away from print.
The Times has been hit by declining print sales and advertising, and said last year it takes in more revenue from readers that from advertising, in a major shift.
The company also sold off the Boston Globe and other regional newspapers to focus on its core operations, and also divested other assets including its stake in an online employment website.
The company brought in Mark Thompson, the former BBC chief, who became president and chief executive at the Times in 2012, as part of its effort to manage a digital transition.
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