“It’s a big day, just on that personal level we’re all incredibly proud of what Kim’s done,” he told the BBC.
“She was incredibly brave to step forward into it, not just around the security side of things given what happened to Jo but also the context, it was a very bruising and pretty horrible campaign at times.
“To put yourself into that and try and keep positive and try and keep a vision on what you want to change and how you bring people together I think she’s done an amazing (job).
“Win or lose frankly we would have been just as proud of her.”
Mr Cox said that the majority of the by-election in Batley and Spen had been good natured but that several marginal candidates had sought to divide communities which was “easy to do in politics”.
He praised Ms Leadbeater for her focus on connecting people, which he said is “ultimately what got her over the line”.
“There’s a differentiation between kinder and still being able to have that lively, energetic, passionate debate and I think the majority of the campaign between the Conservative Party and the Labour Party was that,” he said.
“It was about the ideas, about how you change things for local people, about who’s the best candidate to do that.
“At the margins you have candidates that are focussed on dividing and splitting communities and it’s very easy to divide people, that’s the easy thing to do in politics.
“The difficult thing is, the poetry of politics, is how you bring them together.
“How do you manage to give people that hope and want people to connect with each other and their communities.
“I think that’s what Kim focused on resolutely and I think ultimately that’s what saw her over the line.”
Mr Cox said he had not yet woken up his children to tell them of Ms Leadbeater’s victory but knew that they would be “incredibly excited” about the outcome.
“I haven’t woken them up yet, I’m trying to keep them asleep for as long as possible so their energy levels don’t dip too much during the day but they’re going to be incredibly excited,” he told the BBC.
“We had a call with Kim yesterday and just talking about, win or lose, how proud they were of her.
“They’ll be incredibly excited. I’m just trying to put off the moment when they start bouncing around the house.”
Ms Leadbeater said she hoped to do her sister proud as the new MP for Batley and Spen.
“It was a very big decision to put myself forward. It has been a very emotional campaign and today is very emotional for me for lots of reasons,” she told BBC Breakfast.
“But if I can be half the MP Jo was I know I will do her proud and I will do my family proud.”
Ms Leadbeater’s sister Jo Cox was the Batley and Spen MP when she was murdered in her constituency by a far-right extremist in June 2016 during the EU referendum campaign.