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GB had opened their campaign with a comfortable 31-7 win over Kenya, Dan Bibby (2), Dan Norton, Phil Burgess and Mark Bennett all scoring.
Against Japan - who upset joint favourites New Zealand 14-12 - GB led deep into the final stages.
But Japan scored a third try, only to miss a conversion to draw with time up.
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New Zealand - who will be without Sonny Bill Williams for the rest of the tournament after he partially ruptured an Achilles tendon - bounced back from their shock opening loss to Japan to beat Kenya.
Augustine Pulu scored 11 points, including three conversions, as the All Blacks recovered from conceding a try inside the first minute to win 28-5.
Team GB only just maintained their 100% start to the tournament when they met Japan, James Rodwell scoring twice and Marcus Watson adding another with captain Tom Mitchell converting all three tries.
The Japanese could have drawn after the final hooter when Lomano Lemeki crossed for Japan's third try, but Katsuyuki Sakai missed the tricky drop-goal conversion.
Britain face New Zealand at 16:30 on Wednesday in the final match in Pool C.
The top two sides in each of the three pools go into the quarter-finals, plus the two best third-placed teams.
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Shane Sutton resigned as the body's technical director on Wednesday amid claims of discrimination against elite cyclists, which the Australian denies.
Brailsford, Sutton's predecessor, has promised to co-operate with an inquiry into bullying claims.
"The British Cycling model wasn't based around motivation through fear or performance through fear," he said.
Brailsford, speaking to BBC Radio 4's Today programme, added: "You can't get performance on a continuous basis over a long period of time through fear. I just don't believe it."
Sutton, 58, was made British Cycling technical director in 2014 when Brailsford stepped down after a decade in charge, but is alleged to have used derogatory words like "wobblies" and "gimps" to describe Para-cyclists.
Former European sprint champion Jess Varnish also claimed Sutton made sexist comments towards her and told her to "go and have a baby".
Sutton has said he "rejects the specific claims".
Brailsford has promised to share his knowledge of the workings of British Cycling with an independent review into claims of bullying and discrimination, to be led by Annamarie Phelps, the chair of British Rowing.
"It's really good that this inquiry has been brought in, so we can establish the truth," Brailsford said.
"I'm not going to go into the details. What I will do is I'll share everything that I know with that inquiry."
Great Britain won eight gold medals at the Beijing Olympics in 2008 and matched that total four years later at London 2012, when Sutton was head coach and Brailsford was performance director.
Britain's men made it two wins from two in the Olympic sevens as they edged past New Zealand's conquerors Japan 21-19 in a thriller in their second match.
[NEXT_CONCEPT]
Sir Dave Brailsford has dismissed the idea that British Cycling tried to motivate its riders through fear.