Artemis Racing made the flawless response in the second match, leading from start to finish in a highly tactical race to get off the mark and even the series at 1-1.
Artemis is through to face Team New Zealand in the America's Cup challenger finals.
The win completed a rebound from a 3-1 deficit and represented a big step in Artemis' road back from the tragic death of British Olympic star Andrew "Bart" Simpson during the buildup to the 2013 America's Cup.
"It's two big, well-supported teams going head to head", he said.
"That was obviously the key moment", an elated Outteridge said of the way the race unfolded in high winds and choppy seas. "Before Nath went off, it was shaping up to be a really close top mark, where we'd kind of battled our way back in".
"Words don't describe the feeling when we got back to the dock and saw all the family, team members and friends", continued Barker.
They advance to face the victor between Sweden's Artemis Racing and SoftBank Team Japan.
Team New Zealand won both races against Artemis in the round robins.
If Oracle Team USA or Artemis Racing win, a framework has been agreed among five of the teams for taking it through to the next event, once again in the high-octane foiling catamarans which have proved popular with sailors and broadcasters.
And although Britain's Ben Ainslie, the most successful Olympic sailor ever, counts for almost 1 in five of the medals with his four and one silver, the other America's Cup Olympians have amassed another 22 between them, a Reuters survey shows. "It's going to be a really good race the next couple of days".
Iain Percy, who won two golds and a silver in successive Olympics, said communicating with his heart rate pumping near its maximum while "grinding" the winches is as tough as it gets.
Sponsor SoftBank has said it wants to bring the "Auld Mug" to Japan and teamed up with holders Oracle Team USA, buying a design package to save costs on building its multi-million dollar craft. After the Kiwis reached match point, Oracle won eight straight races to retain the oldest trophy in global sports.
Team New Zealand had just pulled ahead of Artemis when Outteridge had his splashdown. Skipper Russell Coutts handed the wheel to Barker, his understudy, for the clinching fifth race.