The children and their guardians were supposed to leave Poland on Monday, but a vital document from the Ukrainian government was not ready in time.
They will stay in the Callander area, near Stirling, before moving to Edinburgh in small family-style groups.
Steven Carr from Dnipro Kids, which arranged the evacuation, said he was “ecstatic” to get them to safety.
The Edinburgh charity, which was set up by Hibernian fans and has been supporting the orphanages for many years, enabled the children to flee Ukraine for Poland.
First Minister Nicola Sturgeon tweeted a welcome, and wrote: “I know you’d all rather be at home in Ukraine but you’ll find love, care and support here for as long as you need it.”
Dnipro Kids has arranged accommodation for the children, who range in age from one to late teens, as well as two older “sisters” and their seven legal guardians.
They will stay in a rural hostel while they acclimatise to life in Scotland, then move to Edinburgh to live within the smaller household groups they had at home in Dnipro until it is safe for them to return to Ukraine.
Mr Carr told BBC Radio Scotland’s Good Morning Scotland they were “ecstatic” to finally get the children to the “quiet and secluded” hostel where they can be together and the orphanage “mothers” can support each other.
He said: “Dnipro Kids has got a fantastic committee in place that has been working non-stop to make sure everything is in place for the kids when they arrive.
“It’s just great to finally get over the line.”
Although the charity was initially set up by Hibs fans following a UEFA cup match in Dnipro in 2005, he said it had gained “phenomenal” support since beginning the evacuation from Ukraine.
Many people have offered money, time and services, including members of the Ukrainian community in Edinburgh who are helping with language issues.
The children will continue their education with an online Ukrainian programme being used by displaced students during the conflict.
“We just take care of them, we make sure everything is in place,” Mr Carr said.
“It’s very traumatic [for the children] – I wouldn’t say they are excited to be here, hopefully that will happen over the next few days.
“I think at the moment it’s just relief they are now here, they know they are no longer in limbo, they know they can now settle down and get themselves adjusted.”
Last minute visas for ‘sisters’
Last minute visas were arranged on Wednesday for two young women who belonged to one of the orphanages after someone connected to Dnipro Kids offered to sponsor their visas.
The 19 and 21-year-olds thought they would be separated from their “brothers and sisters” and the woman they called mother.
Mr Carr said: “The UK government was brilliant at fast-tracking their visas.”
Daniel Burger, chief executive of Magen David Adom UK helped organise the Virgin flight from Warsaw to London.
He said the flight attendants made sure there was a “celebration” on the flight from Warsaw.
“The plane was full of toys and other goodies and sweets and chocolates and other goodies for them,” he said.
Sally Becker, the founder of Save a Child, was also on the flight from Warsaw on Monday.
“As it touched down they all started to clap,” she said. “The main thing is they’re here, they have sanctuary, temporary sanctuary and they’re safe.”
‘Beginning of something significant’
SNP MP Ian Blackford has been supporting Dnipro Kids deal with the applications for the children to seek sanctuary in Scotland.
He said it had taken time to get the paperwork in order because they had to address all safeguarding concerns before they could be moved to another country.
Speaking in the House of Commons on Wednesday, he thanked the Scottish and UK governments for helping make arrangements for the children.
“This is a good day for these 50 young Ukrainian orphans arrive in Scotland after delay people but let’s hope it’s the beginning of something much more significant for more young people that we can offer sanctuary to,” he said.
Home Secretary Priti Patel tweeted about their arrival: “Huge thanks to my team at the UK Home Office, the Ukraine and Poland authorities, the Scottish Government and Virgin Atlantic who worked urgently on their swift arrival.
“The care they will receive will go some way to heal their suffering.”