Speaking at a press conference, family members said Mr Cochran, 54, and his wife Melissa had "fallen in love" with London during their first trip outside of the US - part of their 25th wedding anniversary.
The couple were both mown down on Westminster Bridge last Wednesday by the car driven by Khalid Masood.
Their family only realised they had been caught up in the attack, however, when scanning photographs from the scene posted onto the internet.
Mr Cochran suffered catastrophic injuries and died later that day, while his wife remains in hospital.
Having arrived in the UK on March 3, the pair had already visited Scotland, Ireland and Stonehenge by the time they arrived in London.
They had been due to return home to the US the day after the terror attack.
Mrs Cochran's mother said the pair had been "loving every minute" of the trip and "couldn't wait" to upload thousands of photos.
She claimed the couple had fallen in love with London and that Mr Cochran repeatedly said he felt "at home" in the capital.
Speaking in London, Clint Payne described his brother-in-law as an "amazing person" whose death has been a "humbling and difficult experience" for the family.
"The most difficult part of this is that Kurt is no longer with us and we miss him terribly," he said.
"He was an amazing individual who loved everyone and tried to make the world a better place.
"He left a legacy of generosity and service that continues to inspire us."
Other victims of the terror attack were on-duty policeman PC Keith Palmer, 48, Leslie Rhodes, 75, and mother-of-two Aysha Frade, 43.