Adam Lewis said the impact of the crash was "overwhelming" but said he was just thankful his family were still alive.
"You don't expect a big white van to come and smack you off the pavement," he told Sky News, adding, "Why someone would ever want to do that I will never know."
Mr Lewis said "our hearts go out" to the family of Karina Menzies, who was killed in a separate attack during the hit-and-run rampage, and admitted: "We are so relieved we're still ok."
Mr Lewis said his "brave" partner will have an operation to put pins in her leg after breaking her fibula and tibula in the crash, while a hairline fracture in her ankle will be strapped. She also wounded her left hip. He added that his two-year-old "had been incredibly brave."
Mother and child were 'catapulted' by rampaging van
Adam Lewis, who saw his partner and baby daughter hit by the rampaging van in Cardiff, has spoken of the moment the vehicle smashed into his family.
With two-year-old Amelia May sat next to him bearing heavy facial scars from the collision, Mr Lewis told Sky News his parter Annie was pushing their daughter's pushchair on the pavement when "all of a sudden we got hit behind by the van."
He added: "It was too late he just smashed right into us."
The father, who was knocked to the floor by the van's wing mirror, said: "Unfortunately my partner and our baby bore the brunt of it."
He said Ms Lewis "buckled" when the vehicle struck and "just flew" with her and Amelia "catapulted" into the air.
The family of Karina Menzies, who was killed in a hit-and-run van attack in Cardiff, have released the following statement:
"Karina was a lovely girl. She lived for her kids and her brothers and sisters and everything she did was for her family. She was a good friend to everyone and was well thought of in the community. She will be sorely missed by everyone."
Hit-and-run victim died 'saving her children's lives'
The brother of the woman killed in a hit-and-run attack in Cardiff said his sister died trying to save the lives of her children.
Craig Menzies described the moment the van careered towards Karina Menzies and her children. He told Radio 5 Live:
He went through the traffic, aimed for them and then just hit them head on. She just screamed and threw them out of the way as much as she could. He still clipped them but she took it head-on. She saved their lives.
Mr Menzies said the attack was "a waste of a life" and said his sister was "the world to me".
She was my best friend. She was a lively, bubbly person. There's just so many good things you could say about her. We'll all be lost without her.
Ely reverend says community is in 'state of carnage'
Reverend Jan Gould, who will lead a service tomorrow for the victims and families of the town's fatal hit-and-run collisions in Ely's Church of the Resurrection, said the feeling in the town was "like a state of carnage".
This will be an opportunity for the whole community to come together, share their grief and find support. People here are numb at the moment and in shock - it's like a state of carnage.
Fortunately, despite its problems, Ely is a very close-knit community and people here are wonderful at pulling together and supporting each other.
– Reverend Jan Gould, the Church of the Resurrection in Ely