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."
Police are investigating a serious sexual assault on a 19-year-old woman in The Meadows area of Edinburgh.