Hamilton, who created the partially-improvised comedy with Guy Jenkin, explained that the broadcaster was unsure about the show because of the failure of Ben Elton's programme Blessed.
According to Chortle, Hamilton told an audience at The Comedy School: "The BBC had just commissioned a thing from Ben Elton about a family with a new baby which died on its arse for all sorts of reasons.
"One is that it's actually a really boring subject - babies are just robots - but when you get young children who ask you questions all the time, that's more interesting."
Hamilton explained that the production company Hat Trick eventually won enough money to make a 15-minute sample of Outnumbered but admitted that the budget was so small that they filmed it at his house.
He added: "To be fair to the BBC, the moment they saw the sample they got it."
Three series of Outnumbered have been broadcast in the UK and an American remake starring Ugly Betty's Ana Ortiz is currently in the works.