"I laid it on very thick, telling him, 'People won't die if you do our sketch'," the Gavin & Stacey star explained. "It makes it very difficult for people to say no.
"He said, 'Bloody hell, James, I've heard some grovelling in my time'.
"I then said, 'That's nothing - if you'd said no, I was going to say I would name my unborn child after you'. To which he said, 'If you promise to do that, I'll do the sketch'.
"And that's why my son is called Max McCartney Kimberley Corden," the comedian tells the Graham Norton Show on Friday night.
James and his wife Julia Carey welcomed their baby boy Max into the world in March 2011, shortly after the sketch aired.
The sketch showed Sir Paul agreeing to film a Comic Relief video in Africa after turning down James, who played his character Smithy from Gavin & Stacey, for being "a bloater".
Paul and James appear on the show on Friday night alongside Katy Perry, Natalie Portman and Chris Hemsworth.
The 71-year-old Let It Be singer also reveals how his relationship with his third wife Nancy Shevell helped inspire his latest music.
"Being loved-up helps," he said. "When I was writing the album in our early days, Nancy was in New York and she was five hours back so I would drop my little girl at school and I'd come home and I'd have a few hours where I didn't have anything on so I would write a song.
"I would then ring Nancy and say, 'Good morning, would you like to hear a song?'. I'd play her the song and it became a really great motivation."
The Graham Norton Show airs on 18 October at 10:35pm on BBC One.