Britain will now face Japan tomorrow afternoon in a game that will decide whether or not they progress from Pool A, but they almost came to grief on day two of the tournament.
Having lost to reigning Paralympic champions the USA yesterday, victory over France was paramount.
But the scores were tied after the first and second quarters before Britain edged the third quarter 15-14 and then pulled away in the final phase to win 57-50 - with Aaron Phipps scoring 30 of his team's goals.
The result keeps them in with a chance of collecting a medal, but other results confirmed once again that the battle for gold will be fierce.
In the day's earlier action, one-man scoring machine Ryley Batt continued on his way as Australia tightened their grip on Group B.
Batt, arguably the most well-known Wheelchair Rugby player in the world, scored 37 goals when Australia opened their campaign by defeating Canada yesterday.
And he was it again today in their second group fixture against Sweden, increasing his goal tally by another 30 as Australia triumphed 60-47 to close in on a semi-final place.
Batt, 23, is Australia's highest-classification player, and he arrived in London with a reputation forged from being named Most Valuable Player (MVP) in the Australian national league, the US national league and the World Championships.
'I was a little nervous,' said Batt, following his latest masterclass.
'I think we played a smarter natural game. We knocked them about a bit.'
Canada, bronze medallists in Beijing four years ago behind the USA and Australia, kept their semi-final hopes alive through a 58-50 victory over Belgium - led by Zak Madell.
'It was a great win for us,' coach Kevin Orr said. 'If we want to be in the medal rounds that was an important win for us.'
In the final game of the night, free-scoring Chuck Aoki scored 20 times to lead the defending champions the USA to a 64-48 victory over Japan.
Australia launch the final round of pool games tomorrow morning when they tackle Belgium, before Canada meet Sweden, Britain face Japan and the USA take on France.