The pair appeared by chance on the show's new 'Ring My Door Bell' segment, which takes viewers live to the homes of studio audience members via their doorbell cameras, and asks neighbours to race to the scene in under 45 seconds to win.
Parents Ashleigh and Matt Glayzer had been watching the popular show when they recognised their neighbour Kim Leech's home on screen and made a mad dash next door with the twins in their arms.
Ashleigh, 28, said: "We're made up. It was exciting to be on TV, and the money will be good for the boys - we think we'll take them somewhere nice, or put it towards their first holiday."
Some viewers were sceptical of the new segment and questioned whether the game was staged, but Ashleigh said she had no idea the Saturday Night Takeaway team was filming outside her door.
"We knew that Kim had tickets to the show and that there was something happening around the local area, but we didn't know what it was," she said. "We could see cameras, but we didn't know what it was to do with, or what show it was for.
"We ran from the house, which is right next door, so we only had to cross the driveway. We could see people running from further down the street, but since we were so close we got there first.
"Some people have asked how we got there with two babies - but obviously we couldn't leave them in the house alone!"
She added: "Identical twins don't run in families, so they're more spontaneous. You don't think that you're going to have twins, so when we told we were having them we were shocked - but after the initial shock wore off, we were very excited.
"It's nice to see them waiting for each other and playing with each other; even at this young age they look out for each other and smile when they see each other."