Nathan Outteridge outfoxed fellow Australian skipper James Spithill as his Swedish America's Cup team Artemis Racing handed defending champions Oracle Team USA their first loss of the regatta.
Land Rover BAR had a bad mark rounding halfway through the race and Groupama Team France went flying past and won by 53 seconds.
Emirates Team New Zealand posted two wins to climb into a tie with Land Rover BAR.
Team New Zealand, who fell to Oracle's mighty comeback in the 2013 America's Cup in San Francisco, were a flawless two-for-two on Sunday, beating Team Japan and Land Rover BAR.
Crews will continue competing in qualifiers over the next two weeks for the right to challenge Team USA for the title in the America's Cup match starting June 17.
Team France opened yesterdays racing with a stunning upset of Artemis Racing, of Sweden.
"It is a very good feeling for the whole team, we have been waiting two years for that feeling", said Franck Cammas after his team's victory in their only race of the day.
The British team won their opening race against Sweden's Artemis Racing on Saturday but their momentum in Bermuda has faltered with four straight defeats - two of them on Sunday - with the latest coming despite a promising start. While Artemis Racing responded with a covering move early, they allowed a split further up the leg.
"We had too many mistakes against the French team in our first race and made life hard for ourselves".
Artemis will face Ainslie's Land Rover BAR in the third race scheduled.
Although the wind was shifty and died slightly towards the end, conditions in Bermuda's Great Sound were once again ideal, with the flat water allowing the boats with their towering "wing" sails to "fly" on their foils.
"We were a bit shocked when the blue light came on", he said.
The Kiwis sacked skipper Dean Barker and replaced him with Burling.
"We had a few tiny issues onboard and I misjudged the entry, but that happens", said Burling on his pre-start error which led to a penalty.
Refusing to rest on their laurels, Emirates Team New Zealand eased over the finish line, one minute and 28 seconds ahead of Land Rover BAR to cap an impressive day on the water.
In a statement, Richard Slater, the chief umpire for the independent America's Cup Race Management, said officials "have had a discussion, we have looked at other evidence, information and data, and I think if we were to go back in time and make that call, we would green that call and not penalize Artemis".