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.

Chris Sane left South Gloucestershire Council days before it emerged the cost of building the North Fringe to Hengrove route had risen £4.6m since November.
Amid months of traffic delays, the construction bill is now £16m higher than forecast two years ago.
Mr Sane resigned on 30 June for "personal reasons", the council said.
"We can confirm that Chris Sane has now left South Gloucestershire Council, with effect from 30 June," a spokeswoman said.
"Alternative arrangements have been put in place to cover this role in the interim."
Earlier this week, a report to the council's cabinet authored by the now former head of transport revealed construction delays were behind the rising costs of the MetroBus route, which runs from Cribbs Causeway to the north of Bristol, through the city centre to Hengrove in south Bristol.
South Gloucestershire will need to spend an extra £9.8m above its original £30.5m contribution to the scheme to cover the increased costs.
The report recommended the extra cash is taken from the highways maintenance budget over the next 10 years.
Commuters have been enduring roadworks associated with the scheme in the city centre, on the M32 and various other locations in South Gloucestershire.

Committee chairwoman Bethan Jenkins said it would "better reflect Welsh life" and AMs were pleased it was being considered by the corporation.
AMs also called for an extra Â£30m to be spent each year on BBC English-language drama and broadcasting about Wales.
The BBC said plans for more spending would be announced "in due course".
The broadcaster has previously conceded aspects of Welsh life had not been "sufficiently captured" by its TV services in Wales.
During its inquiry into the state of the broadcast media industry in Wales, BBC Director General Lord Hall told the Culture, Welsh Language and Communications Committee that putting Wales-based news bulletins on Radio 1 and 2 was being discussed as part of a review.
Ms Jenkins said her committee believed "flagship stations such as Radio 1 and Radio 2 should carry Wales-specific opt-outs for their news programmes to better reflect Welsh life".
AMs were "pleased to hear" this was being seriously considered, she added.
The committee's report said there should be no more funding cuts for S4C until an independent review of the channel had been completed.
It also called on ITV Wales to take a more proactive approach to getting its programmes broadcast across the ITV network.
A BBC spokeswoman said: "We've said that improving how we portray and represent the different nations of the UK - including Wales - on our UK-wide services is a priority for the new Charter period.
"As part of this we've committed to spending more on English-language television programming in Wales and will set out the proposals in due course."
Great Britain's Corie Mapp won gold in the inaugural World Cup Para bobsleigh race.
[NEXT_CONCEPT]
The man in charge of delivering a major part of Bristol's MetroBus network has resigned from his role.
[NEXT_CONCEPT]
Welsh listeners to BBC Radio 1 and 2 should hear news bulletins about Wales at certain times of the day, according to the assembly's culture committee.