The 78-year-old singer spoke to Lennon’s son Sean Ono Lennon, 44, and widow Yoko Ono, 87, for new BBC Radio 2 documentary John Lennon At 80.
McCartney recounted the day on 6 July, 1957, when he was introduced to Lennon, then aged 16, in Liverpool.
He said: “I look back on it now like a fan, how lucky was I to meet this strange teddy boy off the bus, who played music like I did and we get together and boy, we complemented each other!”
McCartney and Lennon went on to form The Beatles with the late George Harrison and Ringo Starr, and they became one of the world’s most famous bands, with 13 studio albums and 17 number one singles.
But the Hey Jude singer admitted that when he and Lennon first started writing songs together, they weren’t all destined to be hits.
McCartney said: “There were a few songs that weren't very good… you know, clearly young songwriters who don't know how to do it.”
He then played an example on his guitar of a Lennon-McCartney track called Just Fun that they never recorded.
McCartney said: “Eventually we started to write slightly better songs and then enjoyed the process of learning together so much that it really took off.”
The Beatles released their last album Let It Be in 1970 before breaking up. Relations were strained between Lennon and McCartney at the time.
But McCartney revealed that while he always believed it was a gloomy time, he saw a picture taken by his late wife Linda that reminded him of the strength of their friendship.
The Beatles star also spoke about Lennon’s insecurities, claiming his bandmate’s confidence was a “shield”.
McCartney said: “Wait a minute, there's this guy 'John Lennon' who's like a genius, clever, witty, confident, and everything, why would he have insecurities? Because we're all fragile beings.”
Lennon was murdered by obsessed fan Mark Chapman outside his New York apartment on 8 December, 1980, aged 40.
Chapman recently apologised for killing the singer.
He told the parole board at New York’s Wende Correctional Facility in August this year: “I just want to reiterate that I’m sorry for my crime. I have no excuse. This was for self-glory. I think it’s the worst crime that there could be to do something to someone that’s innocent.”
He was denied parole.
The two-part documentary will mark what would have been John Lennon’s 80th birthday on Friday, 9 October.
John Lennon At 80 will air on Saturday 3 and Sunday 4 October at 9pm on BBC Radio 2.