In a joint statement from Poppyscotland and Legion Scotland, they said:
“This is a truly deplorable and disrespectful act of malicious violence which has caused immense upset to countless people.
"The Garden and Field of Remembrance is a very special place and thousands of locals and visitors alike spend time reflecting and remembering there each year.
“We are working with Police Scotland to identify the perpetrator of this despicable act.”
Poppyscotland's Head of Fundraising Gordon Michie told Forces News: "When the team arrived at the Garden of Remembrance on Saturday morning, they came across a devastating sight.
"They were naturally quite shocked and dismayed by what they found that morning."
He said that, while initially there were concerns over wind damage, it was soon established it was a result of vandalism:
"We have got a fantastic team of volunteers that have, for many years, manned that gap of remembrance with a lot of great pride, looking after members of the general public that want to come and place their crosses in remembrance of those who have served, many of them who have give the ultimate sacrifice.
"For one particular veteran, one of our older veterans - he fought in the Korean War - it has really hurt him to the core."
Staff worked to restore the site after contacting police.
The garden is a tribute to those who have served or continue to serve in the UK's armed forces.
Hundreds of hours are spent each year planting more than 8,000 poppies.
Wreaths are also laid by representatives of a wide spectrum of organisations during a service to remember the fallen at the opening of the garden.
Inspector Alan Struthers, of Howdenhall police station, said: "This is a reckless and thoughtless act and we need the public's help to trace whoever did this.
"We have already conducted local enquiries. The gates were locked so the person or persons who did this would've had to climb over the fence."
WITNESS APPEAL: Between midnight on Fri 2 Nov and 9am on Sat 3 Nov 2018 the poppy garden on Princes Street Gardens East (near the Scott Monument) was vandalised. If you saw anyone acting suspiciously around that time, contact us on 101 quoting incident number 1318.