The works were among a collection of 1,400 anonymous pictures up for grabs on Friday in the annual sale.
Members of the public gamble on whether they are buying works by celebrity artists at a fraction of their value as a picture's creator is only made known after it has sold.
Other star artists revealed on Friday included Damien Hirst, Julian Opie, best known for designing a Blur album cover, and Turner Prize nominee Keith Tyson.
Potential purchasers queued for hours around the block ahead of the opening of the Secrets sale, where everything costs £35.
The pieces by Sir Paul were bought by different people. One went to a couple who had travelled from Carlisle.
A spokeswoman said the couple had been "shocked" to discover they had bought a work by the singing star.
Child's choice
Second in the queue was Carlos Briones and he bagged the two cards by Opie.
All three Damien Hirst cards sold later. One went to a man from Northern Ireland.
Another was chosen by a two-year-old girl who was with her father.
The spokeswoman said all had been "surprised and delighted".
Other famous contributors to the sale included clothes designers Stella McCartney - Sir Paul's daughter - Giorgio Armani, shoemaker Manolo Blahnik and furniture designer and restaurant owner Sir Terence Conran.
The sale continues until Sunday but the spokeswoman said the cards were going so fast things could wind up early.
Bowie
The Secrets sale is one of the highlights of the British art world's calendar and always creates a great deal of interest.
Last year one woman at the head of the queue had spent four days camping to ensure she got her hands on her favourite postcard.
The Secrets show is sponsored by David Bowie's company Bowieart.
The money from the sale goes towards the Royal College of Art's Fine Art Student Award Fund, which helps support emerging student artists.
Much of the exhibition is made up of works created by students at the college.