America's Cup action was postponed due to high winds on Wednesday with all four challenger play-off semi-final races moved back to Thursday in Bermuda.
Britain, meanwhile, still hasn't bounced back from its loss in 1851 to the schooner America.
They advance to face the victor between Sweden's Artemis Racing and SoftBank Team Japan.
But after making a crucial error in the first race and allowing New Zealand to get past them, Artemis Racing sailed a near ideal race in their second encounter to level the series.
Barker was skipper when Team NZ squandered an 8-1 lead to lose the Cup Match 9-8 to Team USA off San Francisco four years ago.
The victor of the challenger finals, which start Saturday, will face Oracle Team USA in the 35th America's Cup match starting June 17.
"When something goes wrong like capsizing you didn't see people crowing about it or blaming Peter Burling or his crew because we know that could just as easily be us tomorrow, that's how hard you need to push these boats and mistakes will happen". "Full credit to them". After the Kiwis reached match point, Oracle won eight straight races to retain the oldest trophy in worldwide sports.
New Zealand's hotshot Peter Burling, the youngest helmsman in the event at 26, and British tactician Giles Scott, 29, both jumped straight back into the America's Cup boats after winning gold medals in Rio previous year.
Burling said the capsize will be in the back of his mind for a long time. Artemis Racing have their base at the other end of the Sound at Morgan's Point and it is assumed that they sailed in the designated, but smaller Challenger race area. "We will take that opportunity to keep working on the boat". "It was a great day for us to be able to repay the shore crew by getting it done with a day to go".
Ainslie, who had crossed the start line in front, immediately slowed and instructed the support boat following him to return to render assistance.
The Kiwis then saluted the British crew.
Devoid of their captain, the Swedes gave up the ghost in Bermuda's Great Sound and retired, allowing the Kiwis to freely sail to victory.
"We can simplify it and try and hold this every two years".
"We have been very fortunate with our relationship with Oracle", Barker said.
"We're absolutely stoked", Outteridge said shortly after clinching the win.
"We are really excited to get through to this next stage".
Mark Cameron, Land Rover's Experiential Marketing Director, added: "The Land Rover BAR team has accomplished so much and we are immensely proud of the partnership that has been forged".
"This is the most successful Japanese America's Cup challenge in history", said Sofuku. He also has a silver medal.
While Dean Barker sailed an nearly faultless race, Artemis Racing did not make the mistake the Japanese team would have been hoping for and Nathan Outteridge steered his team home with a 28 second victory, levelling the scores at 3-3 ahead of the teams' third and final battle of the day.
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