Trump had previously denied all knowledge of the payment to Daniels, before admitting last week that Cohen struck a "deal" with the porn star on his behalf.
Giuliani, who has joined Trump's legal team in recent weeks, made the statement during an interview with Sean Hannity on the Fox News Channel late Wednesday.
He also said the payment was "going to turn out to be perfectly legal" because "that money was not campaign money".
The sum was wired to Daniels just days before the November 2016 presidential election, and there had been suggestions Cohen could have fallen foul of campaign finance laws because it represented a contribution aimed at preventing negative press from arising.
Giuliani said Trump didn't know the specifics of the transfer but knew it was happening. "Michael would take care of things like this". He said Americans should be outraged by Giuliani's admission.
Daniels on Monday filed another lawsuit against Trump accusing him of defamation for an April 19 tweet in which he suggested that she had fabricated story about being threatened in a Las Vegas parking lot in 2011 by a man who told her: "Leave Trump alone".
Cohen and White House aides have denied that Trump had an affair with Daniels.
"I love Rudy, but they better have an explanation for that", she added.
Mr Cohen has said he paid Ms Daniels out of his own pocket and was not reimbursed by Mr Trump. "These are busy people".
Daniels' lawyer Michael Avenatti reacted swiftly to the revelation, calling out the United States president over his "lies".
The 39-year-old - real name Stephanie Clifford - says she had unprotected sex with Mr Trump after he invited her to his hotel room, then pulled his pants down.
"He did know about the general arrangement that Michael would take care of things like this", he said when he revealedPresident Trump's agreement with Cohen.
"Ivanka Trump? I think I would get on my charger and go ride into their offices with a lance, if they go after Ivanka", the former mayor continued. Giuliani told Costa that he and the president had discussed the revelation about the reimbursement of Cohen "long in advance".
Prior to Comey's book release, Trump even insinuated Comey should go to jail, tweeting: "The big questions in Comey's badly reviewed book aren't answered like, how come he gave up Classified Information (jail), why did he lie to Congress (jail), why did the DNC refuse to give Server to the Federal Bureau of Investigation (why didn't they TAKE it), why the phony memos, McCabe's $700,000 & more?".
Mosque blasts kill dozens in northeast Nigeria
The jihadist group on Thursday carried out an attack in Maiduguri, capital of neighbouring Borno state, that killed four people. He detonated his explosives "five minutes before the prayer started", he added. "This is obviously the work of Boko Haram".
Cyber Monday 2017 – The top sellers in electronics this year
Apple’s 9.7-inch Wi-Fi enabled iPad Pro was also still a great seller thanks to massive price discounts from retailers. Microsoft dropped their game changing console update, the Microsoft Xbox One X with a 1TB hard drive and 4K compatibility.