Lance Corporal Mapp, of the Household Cavalry, finished 0.18 seconds ahead of American Jason Storm in Igls, Austria.
"What a feeling," said Mapp. "Delighted to win the first Para bobsleigh gold."
Mapp lost both legs when his military vehicle ran over an explosive device in Afghanistan in 2010. He is in the GB sitting volleyball team and took part in the Invictus Games.
He said: "Being part of the world's first ever Para bobsleigh race today was incredible. To win was something really special.
"I hope that I have done people proud - I've had so much support from many different people.
"Help for Heroes have been instrumental in supporting me to this point - it's been great to have their backing.
"I love bobsleigh. I only tried it for the first time in Calgary last season so to be here in Austria, at the first ever race and win is just amazing.
"I feel so excited about where the sport is going and for our next race in St. Moritz next week.
"The field of athletes has been strong and there's been some tough competition. It was an amazingly close race - only three-tenths of a second between the top four finishers.
"To think I nearly quit bobsleigh last summer to focus on my other sports. It's been a fantastic experience to be here, with such a great group of competitors. "
Para bobsleigh and Para skeleton are being included at World Cup events for the first time this season but the International Paralympic Committee have said that the sports will not be included in a Winter Paralympics until 2022 at the earliest.
Great Britain's Corie Mapp won gold in the inaugural World Cup Para bobsleigh race.