The incident happened Thursday evening during the back stretch of the event's fourth heat.
A necropsy done Friday determined that the first horse died of a ruptured aortic aneurysm, said Stampede spokesman Doug Fraser.
Two other horses had to be euthanized on the track due to the seriousness of their injuries, and a fourth was badly injured but is expected to survive after undergoing surgery.
The wagon's driver, Chad Harden, was visibly shaken up by the crash and appeared to be on the verge of tears when he spoke to reporters.
"They're just like humans, they're our family," he said of the horses. "It's hard to take."
Changes sought in how the rodeo operates
While horse deaths at the Stampede are nothing new, the latest incident has renewed calls from animal welfare groups to make fundamental changes in how the rodeo operates.
The Vancouver Humane Society (VHS) has been one of the Stampede's most vocal critics, noting that more than 50 horses have died during the annual rodeo over the past 26 years.
Stampede organizers introduced changes last year in the hopes of making the rodeo safer, after six horses died in 2010, including two from heart attacks.
Under the new rules all horses are inspected by veterinarians when they arrive at the Stampede, and before and after every race. Horses are also given a day of rest after four days of racing.
To make the races less crowded, organizers reduced the number of outriders running alongside each chuckwagon from four to two.
However, animal rights groups say the changes have failed to stop horses from incurring fatal injuries.
"We're extremely frustrated because horses have been dying almost every year in the chuckwagon race," said Peter Fricker, a spokesperson for the VHS, which is calling for the race to be suspended pending an independent safety review.
"It may be that there isn't any way of making it safe, but a review needs to look at what exactly is going wrong."
The problem could lie with the type of horses that are selected to compete, Fricker said.
He pointed to research by Temple Grandin, an internationally renowned animal behaviour expert in the United States who has suggested that over-selective breeding may have weakened the legs of thoroughbreds, including those used in chuckwagon races.
Comparing show jumping and rodeo
And it appears that other types of competitions may have a much lower occurrence of horse fatalities.
Equine Canada, the governing body for equestrian sports like show jumping, has held about 1,800 competitions over the past three years, each of which can feature hundreds of horses. During that time there have been three deaths during competition, said Dr. Mary Bell, chair of the organization's health and welfare committee.
By contrast, Fricker said it's nearly impossible to find reliable statistics for horse fatalities on the wider rodeo circuit across the country.
"But it's safe to say if there are animals dying at the Stampede, there are animals dying at other rodeos across Canada."