The woman had been out with friends in the Cowgate and Niddry Street area and was returning home at about 07:00 when she was attacked near Jawbone Walk.
Police said she had been making her way through Middle Meadow Walk between 06:50 and 07:10 when she was approached by a man from behind.
He knocked her to the ground before carrying out the "horrific" attack.
The man, who was white, aged 20-30, 5ft 4in to 5ft 9in tall, of medium build. He was wearing a black zip-up top, dark trousers and a black beanie hat.
He made off before a member of the public came to the woman's aid.
Police have appealed for anyone with information to come forward.
Det Ch Insp Alwyn Bell said: "This has been a horrific attack on a young woman and she is understandably extremely distressed at this time.
"Specially trained officers are currently working alongside our relevant partner agencies to offer the victim all the necessary support and assistance she requires.
"This incident took place in an area that is often busy with joggers, dog walkers and other members of the public, regardless of the time, and I would urge anyone who was in the Meadows on Saturday morning and remembers seeing anything suspicious to contact police immediately."
He added: "We will have high-visibility police patrols deployed within the Meadows for the time being to offer reassurance to the public and to engage with anyone who may have information relevant to this inquiry."

The raids destroyed nine militant positions in the Tirah Valley in Khyber district, the military said.
The strikes came after the Taliban stormed Karachi airport, in an attack that killed at least 30 people.
The Pakistani Taliban said Sunday's assault was in revenge for the killing of their leader last year.
"Nine terrorist hideouts were destroyed by early morning military air strikes near the Pakistan-Afghan border," the military said in a statement on Tuesday.
The Khyber tribal region, near the Pakistan-Afghan border, is believed to be a base for several militant groups and foreign fighters.
It is not clear when the air strikes took place or which militant group was targeted, the BBC's Shahzeb Jillani in Islamabad reports.
Late on Sunday, 10 heavily armed Taliban fighters attacked an area of Karachi's Jinnah international airport used mainly for cargo and private flights.
Security forces gained control in the early hours of Monday.
The airport reopened late on Monday.
Officials said at least 29 people were killed in the fighting which raged from Sunday into Monday, including all the militants. On Tuesday, officials said nine more bodies had been recovered from the airport.
The number included seven bodies found in the airport's cold storage facility. Officials said they were charred beyond recognition.
There were reports that the bodies belonged to a group of airport employees who were trapped in the facility after taking refuge from the attack.
Pakistan has been fighting an Islamist insurgency for more than a decade, with the Pakistani Taliban the main militant grouping.
The airport attack came against the backdrop of a major split in the Pakistani Taliban, and threats of retaliation following limited military operations against foreign militants in North Waziristan.
The brazen attack has brought the government of the prime minister, Nawaz Sharif, under renewed pressure to order tough action against the Taliban militants, says the BBC's Shahzeb Jillani.
The Pakistan government began peace talks with the Pakistani Taliban in March, but little progress has been made and the violence has continued, with Karachi a frequent target.
Correspondents say that given the violence, it seems clear that any pretence at a peace process is now over.
Police are investigating a serious sexual assault on a 19-year-old woman in The Meadows area of Edinburgh.
[NEXT_CONCEPT]
The Pakistani military has carried out air strikes in tribal areas in the north-west of the country, killing at least 15 militants, officials say.