He said: "It was really touching, and what a privilege.
"We never thought we'd see these episodes again. It was a real factory process, no time for post-production or anything like that, never repeated, never sold abroad.
"People will be thrilled. The thing about Doctor Who is it's a connection to your childhood.
"I'm a grown man and I've been wishing I could see The Nightmare Begins since I saw the name on a list of missing episodes of Doctor Who 30 years ago.
"Forget Glastonbury, I think if you put on a screening of these tomorrow it would sell out in seconds.
"They really are the holy grail."
A special screening of the two episodes organised by FIF, with Purves as guest of honour, is being held in London on 4 April.
Details of the event are yet to be announced, but it will take place on the same day the two restored episodes are made available in full on BBC iPlayer.
Episode one sees the Tardis landing on the planet Kembel in search of help for the Doctor's injured companion, only to learn the Daleks are planning with Guardian of the Solar System Mavic Chen to conquer the Earth.
By episode three, the Doctor has stolen a vital component needed for the conquest, but after leaving Kembel they are forced to land on penal planet Desperus - pursued by the Daleks.
Noreen Adams, director of BBC Archives, said she was "thrilled" the episodes would be released on BBC iPlayer.
She said: "BBC Archives has been working to restore the original recordings and update these to broadcast quality, ensuring fans can enjoy a little extra treat with their Easter Eggs this April."