Groupama Team France is led by Frenchman Franck Cammas.
Ainslie was held responsible by race umpires for causing the collision, incurring a penalty, which meant the team had no recourse to redress. However, despite Nathan Outteridge's team chasing down their French rivals, Groupama Team France kept their nerve and clinched an impressive victory.
"We finally figured it out and that was it -check mate", said skipper Jimmy Spithill.
Struggling Land Rover BAR has three points, followed by Artemis, SoftBank Team Japan and Groupama Team France with two points apiece.
It was the British team who started the stronger. We learned more today than we've learned since we've been here and probably since we launched this boat.
Artemis Racing remained in hot pursuit throughout, but unlike in race 2 against SoftBank Team Japan, they could not stage a fightback and Land Rover BAR remained composed, foiling impressively throughout their manoeuvres.
"Our boat is pretty badly damaged", Ainslie said.
Also taking part in the competition in Bermuda are teams from Sweden, New Zealand, Japan and France, who will all be sailing at up to 60mph.
But they were lucky that they didn't take the boat or some of the Japanese team with them, such was the velocity the two boats were travelling at during the race.
'We would have liked to have won more races, but I'm not scared or anxious, ' said Land Rover BAR team manager Jono Macbeth.
Earlier, Burling gained an important edge over predecessor Dean Barker, with victory over Team Japan in their first meeting of the qualifier series.
But Ben Ainslie's crew then lost to SoftBank Team Japan in the final race of the day after a spectacular high-speed collision between the two catamarans, which have taken huge design teams and millions of dollars to develop.
Commenting on the race against Artemis, Sir Ben said: "We had a fantastic race against Artemis Racing". The race of the day is expected to be the second between Artemis Racing and Emirates Team New Zealand. But the decision stands and here's how helmsman Peter Burling and skipper Glenn Ashby set up the doubt in the on-course adjudicating. However, despite the lapse in concentration and having to start two boat lengths behind the British team, the incident didn't prove disastrous as they bounced back spectacularly.
"When you look at the conditions, it's no surprise at all that either team could have been a long way ahead", he said. I think it has been [advantageous]; staying here and training here certainly helps the team. "I obviously have a bit of a connection [to the USA] because my wife is American and my two boys were born in America, but I'm really happy with this team, a team with a lot of nationalities, and we're proud to be representing the States".
America's Cup Qualifiers: From May 27 to June 3, all six America's Cup teams - yes, that includes the Defender - will compete in a double round robin match race series where each competitor races every other team twice.
"When we stopped using the conventional sail, we went to the big carbon-fibre rigid wing, I just figured the best way to learn about aerodynamics and wings was to get a [flying] licence", he explains.
"It was a challenging day, but we had a couple of legs where we were able to thread the needle and make some good gains".