Matavao crossed with eight minutes left in Canberra to level the scores and Pat Pellegrini added the conversion to ensure the cash-strapped Auckland-based side ended their campaign on a high.
Defeat left the Brumbies sixth on the ladder and a difficult play-off game ahead against the table-topping Wellington Hurricanes.
Established in 2020 to provide a top-level pathway for players of Pacific heritage, Moana finish rooted to the bottom of the table and with major doubts over their future.
The team's ownership group, Pasifika Medical Association, announced in April it would cease its funding at the end of the season after failing to turn them into a sustainable commercial enterprise.
Other parties are reportedly interested in acquiring the licence, but nothing concrete has been put forward and, without an 11th-hour rescue, the clash will be their last.
Players were emotional after the game, some shedding tears.
"A pretty emotional week and I'm proud. It's been a privilege to lead these boys," Moana skipper Miracle Faiilagi said.
"It was an 80-minute performance today, that's what we want, staying in the fight. To come here and win is quite special for us.
"We had something to play for and this win means a lot to us. This is for our people."
Should Moana go out of business, the competition will be down to 10 teams next year after the Melbourne Rebels similarly shut down because of financial difficulties in 2024.
Even if they are saved, there are questions around their squad with many current players already signing contracts elsewhere.
"You've got to give credit to them, they are a passionate side," said Brumbies captain Allan Alaalatoa.
"We hope this isn't the last we see of Moana and obviously today it was a great performance by them."
The Brumbies were heavily favoured and it looked like a rout could be on the cards when Rory Scott crossed in the second minute and then Tom Wright added another try.
But Moana showed plenty of fight and hit back with a converted brace from Pellegrini to level the scores 14-14 at the break.
A try for Luke Reimer restored the hosts' lead soon after the restart and their hopes appeared to be over when Moana centre Faletoi Peni was red-carded for a dangerous tackle.
But they never gave up, battling until the end for a memorable win despite playing the last 18 minutes one man down.