Sophie Thornhill and pilot Rachel James clocked one minute 5.912 seconds to win the women's race in Aguascalientes.
Neil Fachie and Pete Mitchell won the men's event in 59.460 seconds - the first sub-one minute tandem kilo.
Jon-Allan Butterworth claimed bronze in the C5 4km pursuit to take the GB medal tally in Mexico to four.
Sarah Storey won bronze in the C5 500m time-trial on the first day of the competition, which finishes on Sunday.
Thornhill, 18, and James, the sister of able-bodied world champion Becky, only came together as a pairing late last year and were riding in their debut race at a major international event.
Double gold for GB para-cyclistsThey beat the record of 1:08.714 to finish 1.608 seconds clear of Australians Brandie O'Connor and Breanna Hargrave.
"To think this is our first race at a major championships, and we won it, I still can't believe it," Thornhill, from Cheshire, told BBC Sport.
"We knew the record was on the cards but to break it by almost three seconds is amazing."
Aberdeen rider Fachie and Mitchell smashed the record of 1:01.351 which Fachie set in winning Paralympic gold with Sarah's husband Barney Storey at London 2012.
Fachie, 30, told BBC Sport: "I always said when I started cycling it was one of the golden barriers that I wanted to be the first tandem to go under a minute.
"To do it here is very special. It's going to be pretty hard to break that mark now. It's something we can treasure forever."
Mitchell, 24, added: "It's definitely the toughest podium I've done [due to the altitude and heat in the velodrome]. I don't think I smiled for about half an hour afterwards, but it's all worth it."
