They stretched that lead upwind on the next beat and simply had to defend to the finish.
But as they rounded the final mark, the Kiwis fell off their foils and stopped dead, allowing Sweden to close at an alarming rate and almost snatch the point.
"We definitely did it a little tough there, and it just shows one little mistake, one little error and you're really struggling," said Burling.
"It was probably poor communication on my behalf - we all thought we'd be able to lay through the gate and do a nice gybe after the gate, but then, because we didn't lay through, it became pretty hard.
"The boys dug pretty deep - the legs were a little tired at the end of that, but the amount of power they were able to put out to get that board down was pretty key."
In the second race, Artemis were forced to withdraw with a breakdown, but they seemed to successfully fix that fault and there was no sign of any trouble through the next outing.
Burling had chosen to split upwind on the third leg, converged on even terms with his rivals and a bad tack from Artemis allowed TNZ to build an 18-second margin by the fourth mark.
"With our configuration, we always knew if we were at the bottom mark in good shape, we'd have a good crack at them up the beat," said Burling afterwards.
"They did very similar things to what they'd done in the previous race, we just reacted a little better than we have in the past."
From there, it was simply a matter of defending and making no mistakes.
The teams had again gone for very different configuration on their foils, flip-flopping from their choices the previous days. New Zealand had prepared for lighter conditions with longer boards and seemed better suited as the winds dropped for the second race.
"Up that beat, we felt how much better our tacks were," said Burling. "Every time we went through a tack and we got halfway decent shift, we just started sailing away or gaining."
In the day's first race, Team Sweden reached speeds of up to 45 knots to power over the top of the Kiwis up the first leg.
From that point, Artemis simply sailed away from Team NZ, stretching to 20 seconds at the fourth mark.
They had a scary moment up the fifth leg, tacking and then nosediving into the Great Sound, losing much of their margin to the chasing Kiwis, who closed to within five seconds at the next rounding.
As Outteridge lost control and veered towards them, Team NZ had to take evasive action and protested, but their appeal was declined. That was as close as they got to making a pass.
"We just like to keep it interesting for everyone," said Outteridge. "It was just a little bit of a miscalculation with the tack."
Sweden simply had too much boat speed and quickly recovered their poise downwind for a 15-second victory.
"We thought we got a really good start, but they just managed to push that boat that much harder down the reach," reflected Burling.
"We've got a lot to review, but we thought it was touch and go, when they had that little skid up the beat and we had to bear away to make sure we didn't have an incident."
The result means New Zealand cannot wrap up the final series today - we'll be back tomorrow to find who will take on holders Oracle Team USA for the Auld Mug next week.
Today's schedule
05:12 Race 4: Sweden beat New Zealand by 15 secs (2-2)
05:51 Race 5: New Zealand beat Sweden withdrew (NZ 3-2)
06:30 Race 6: New Zealand beat Sweden by 1 sec (NZ 4-2)
