Manu scored a second-half double, including a 74th-minute effort when he dragged two Broncos over the line before finishing the game with eight tackle busts and two linebreaks in an all-action display.
Sam Walker, who shunned the Broncos to start his NRL career at Bondi, iced the game with a crucial conversion from the touchline before then setting up a try to Sitili Tupouniua which proved decisive in the 76th-minute.
Brisbane, after a week of off-field turmoil following a video of teammates Payne Haas and Albert Kelly fighting at a Sydney hotel emerged on social media, have now lost three in a row despite Staggs' stellar display.
Corey Oates' completed his hat-trick with less than two minutes to play as the Broncos tried to snatch the win but Adam Reynolds' missed conversion proved costly as the Broncos received a penalty with seven seconds to play but had to go for a hail-mary try.
The drama-packed finish came after a first half dominated by Staggs.
After Oates had opened the scoring in the 19th minute, Staggs stole the show with a brilliant individual try in the 35th.
The Brisbane centre shrugged off Paul Momirovski before crashing through Walker and going over to have the home crowd on their feet.
Staggs ended the game with 182 running metres and 13 tackle busts in a great performance on a beaten team.
Manu took over in the second half as the Roosters mounted a comeback, wiping out their 10-point deficit within 14 minutes through tries to Walker and Manu.
Oates' second put the hosts back in front before Manu and then Tupouniua's late efforts got the visitors home, despite the dramatic late finish by the hosts.
The Tricolours lost Nat Butcher early in the second half to a HIA after his head accidentally collided with Haas's knee just when it looked like the Rooster was going to scored.
A pair of Roosters were put on report with Butcher cited for a first-half crusher tackle on Ryan James, while Jared Waerea-Hargreaves was penalised for dangerous contact after he twisted James' knee in the ruck.
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