The wins give New Zealand a 2-0 lead in the first-to-seven series, with co-helms Peter Burling and Nathan Outteridge delivering clinical performances in light Mediterranean breezes.
Britain had a last-minute scare just before the start of their first race when one of the batteries which power the boat's foils had "a bit of a meltdown" and had to be removed "in a bit of a hurry", their skipper Ben Ainslie said.
The incident meant the British were slightly on the back foot coming into the first pre-start, with the New Zealanders steaming over the line and getting a jump on the first leg.
"Awesome to get a couple of wins on the board. A really tricky day. So shifty, dynamic, up-down. But really pleased with the way the group managed to make more good decisions than not," said Burling, 33, after the races.
"We're really happy with the way that we went out and performed and we know that we are going to have however many tough races from here," added Burling.
Ainslie's team have qualified as challengers for the "Auld Mug", which began in 1851, for the first time in 60 years.
"A tough day. It wasn't the start we were looking for but some positives ... Whilst they had a little edge at times it was close and certainly that second race showed we can give them a good fight," Ainslie said back ashore.
"There's plenty we'll take away in terms of improvements to make and it's a long series ahead," he told reporters.
Data from Saturday's races will be closely analysed by both teams as they seek to find ways to eke out more speed.
An old adage among America's Cup followers is that "the fastest boat wins", with both high-tech AC75 foiling monohulls set to be put through their paces in the week ahead.
"The difference in boat speed definitely gets exaggerated when you can get on some (wind) shifts as the leader," Burling said when asked about the relative performance of the boats.
The 37th Cup is being contested off the Spanish city of Barcelona, where variable winds and unstable sea states in the qualifying events have made sailing conditions challenging.
Also on Saturday, Italy won the first America's Cup for women, beating Britain in the final. Italy also won the Youth America's Cup in the smaller AC40s used for the women's event.
'BRING THE CUP HOME'
Ainslie's decade-long campaign to "bring the Cup home" has been backed by British billionaire Jim Ratcliffe, who was on the water to see the historic encounter, and they have worked closely with the Mercedes Formula One team to come up with a race-winning package and end "173 years of hurt".
Britain battled their way to challenge New Zealand by beating Italy 7-4 in the Louis Vuitton Cup, with the team showing improved speed and handling through the series.
Earlier, there was a party vibe at the British base in Barcelona, with horns blaring and fans waving flags as the sailors and their support crew docked out.
"We're so grateful to everyone back home and around the world that has got behind the team. We know it's been a long time since we've been in the America's Cup so to see what it means to everyone is just brilliant for the team, it really means a lot," Ainslie said of the support.
A sea of New Zealand supporters in black team kit shirts, many draped in the country's flag, cheered the Kiwis from the dock with a rousing Maori send-off before their boat was escorted towards the sea by a traditional waka.
