Trump announced the news with Acosta by his side at the White House as the president left on a trip to the Midwest.
Update 11:00 a.m. EST: Trump stands with Acosta, says, "It was him, not me".
Trump, who publicly faults the news media almost every day, said Acosta put the blame for his departure there, too.
Acosta becomes the latest in a long line of high-profile officials to resign their post.
Acosta's resignation will go into effect next Friday.
During his Senate confirmation hearing in March 2017, Acosta defended his record as a US attorney and addressed his decision to not indict Epstein on federal charges.
President Trump told reporters Friday morning that Acosta had chose to step aside.
Mr Acosta was the United States attorney in Miami when he oversaw a 2008 non-prosecution agreement with Epstein.
On Wednesday Acosta held a news conference to defend his handling of the deal, which allowed Epstein to plead guilty to a state charge and not face federal prosecution. He paid settlements to victims and is a registered sex offender.
The 2008 plea deal in Florida came under renewed scrutiny in light of Epstein's indictment Monday on more child sex trafficking charges in NY.
From 2005 through 2009 he was the USA attorney for the Southern District of Florida.
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While some within the NHS welcomed the news , others have raised concerns around patient safety and data protection. Amazon has sought to allay concerns from privacy groups that it can not be trusted with such sensitive data.
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Still, Alamilla said, he made a decision to accept the money - but also ban his charity's child clients from attending. The charity's website says it is "a non-profit organization under the laws of the State of Florida".
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Acosta recently said taking Epstein's 2008 case to trail back then would have been a "roll of the dice" scenario, per the Miami Herald.
Acosta was further criticised for leading efforts to dramatically cut labor department budgets dedicated to combating human trafficking. In a 50-plus-minute lawyerly rebuttal, he argued his office had secured the best deal it could and had worked in the best interests of Epstein's victims.
"We have an fantastic economy and the focus needs to be on the economy job creation", said Acosta on July 12 as he stood alongside Trump at the White House. "We believe that we proceeded appropriately".
Mr Acosta had been under pressure from Democrats to quit. "He explained it. He made a deal people were happy with ... now they're not", Trump said from the lawn. "Today's world treats victims very, very differently".
Last week Epstein was arrested by federal law enforcement authorities and charged with a slew of crimes, including sex trafficking of underage girls.
Acosta had twice defended the deal this week, claiming that it was necessary to bring Epstein to justice at the time.
Epstein, 66, was arrested on July 6 at Teterboro Airport in New Jersey, where he had returned on his private plane from Paris. The records also show that Epstein has a net worth of more than $500 million and makes at least $10 million per year, the filing said. "Terrific guy. He's a lot of fun to be with".
In a 2002 magazine profile, he said: "I invest in people, be it politics or science".
Trump has since tried to distance himself from Epstein.
CNN's Chris Cuomo also dressed Ryan down Thursday night, for only slightly different reasons.
"While the wheels of justice will grind slowly for the survivors of this monster Jeffrey Epstein's heinous actions, they clearly moved today". Earlier this week, Mr Trump told reporters that he felt badly for his labour secretary, but added that he did not know him personally before hiring him. He was confirmed in the Senate 60-38.
"Instead of prosecuting a predator and serial sex trafficker of children, Acosta chose to let him off easy", he said.