The Lionesses' third-place finish in Canada makes them the second most successful England team ever, only behind the men's victory in 1966.
Sampson replaced former boss Hope Powell in August 2013.
"It's been instilled in us over the last 18 months, that family atmosphere, the togetherness," said White.
"It's 23 players, we're all in it. You can't win it without 23 players or the backroom staff as well."
During Powell's 15-year reign as manager, the Lionesses never progressed beyond the World Cup quarter-finals - and former players have since criticised the negative feeling among the team during her time in charge.
However, speaking to BBC Radio Three Counties, 26-year-old White says it has been a different story under Sampson.
"He's changed a lot in the team, he's brought in new people," said White. "I always think it's good to have a change.
"He's definitely brought that togetherness, that fire, that passion. Everyone was on board with everything.
"We all stuck together and we're definitely like a little family."
Notts County striker Ellen White says England's success at the Women's World Cup was down to the "family atmosphere" instilled by manager Mark Sampson.