Barrichello and new world champion Jenson Button dominated the early part of this year's title race with Brawn.
Button, who won the 2009 title with Brawn, will drive for McLaren in 2010, alongside fellow Briton and 2008 champion Hamilton.
Button was world champion with Brawn in 2009, and finished runner-up in the drivers' standings last year -- trailing Vettel by 122 points.
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Schumacher's return reforms the partnership with Brawn that brought him all seven of his world titles, the first two with Benetton and the rest with Ferrari.
BBC: Ross Brawn says top Formula 1 teams are closely matched
Webber was punished for colliding with Brawn's Rubens Barrichello at the start but was in a class of his own to dominate the race despite his handicap.
But that was no surprise as the 20 drivers grew accustomed to the track, culminating in the Finn ending up quicker than his McLaren team-mate by just 0.197 seconds, with Brawn GP's Button 0.234 seconds down on Kovalainen.
Button's team-mate Rubens Barrichello endured an incident-packed race but was impressed with the Brawn's durability.
The continentals, with less brawn, more flair, have brought new life.
Briton Jenson Button led home Rubens Barrichello barely a day after Branson confirmed a "substantial sponsorship" deal with the Ross Brawn-owned team.
The 40-year-old last raced in Formula 1 in 2006, but he has been strongly linked with a return to the sport with the renamed Brawn GP team.
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Vitantonio Liuzzi earned two points for the Mercedes-powered Force India team with ninth, and former Brawn driver Rubens Barrichello took the final point with 10th on debut for Williams.
The British GP has been held every year since 1948 and has a rich history, while recently English drivers Lewis Hamilton and Jenson Button have won 2008 and 2009 world titles respectively with Brackley-based Brawn also clinching this year's constructors' crown.
Did you get these things done through sheer force of will, with your bare hands and brawn?
Brawn insisted starting with last year's car was "the right decision, although at the beginning of the season it will be tough".
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"I am pleased with the progress that the team has made over the last month and the steps forward that have been made with the car, " Brawn told reporters.
Formula 1's governing body and Fota have become embroiled in an increasingly bitter feud over Mosley's presidency and his plans for the 2010 season, which include budgetary and technical changes that have been met with opposition by Ferrari, Brawn GP, McLaren, Red Bull, BMW Sauber, Toyota, Renault and Toro Rosso.
Brawn returns to work with Ferrari on 5 January for a week's testing in Jerez.
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One reason is that conservative brawn has lost patience with brains of all kinds, conservative or liberal.
Mercedes team principal Ross Brawn, who worked with Schumacher at Ferrari, conceded he has not provided him with an adequate car.
"We have quite a lot of new parts on the cars this weekend and it looks like there may be something there with the floor, " said Brawn.
Brawn has been impressed with Hamilton during preseason testing, despite a hydraulic problem causing the British driver to crash out of his first drive for the team at Jerez.
Championship leaders Brawn arrived in Budapest with a major upgrade package which includes a new diffuser, rear wing and bodywork and they used Friday's first session to test their new parts with race fuel loads.
Alonso's quickest lap lifted him clear of second-placed Sebastien Buemi of Toro Rosso with local hero Rubens Barrichello of Brawn GP third -- as he aims for the victory that will keep his slim world title hopes alive.
Jenson Button, who has moved to McLaren for 2010, laid the foundations for his world title by winning six of the first seven races, but Brawn were then faced with a rearguard battle for the rest of the season.
BBC: Ross Brawn says top Formula 1 teams are closely matched
Team boss Ross Brawn sealed their survival with a management buyout in early March, ensuring most employees initially stayed to help rebuild a fresh team after the under-performing Japanese outfit pulled out of the sport because of the global financial crisis.
Red Bull looked set to begin the season in spectacular style at the Australian Grand Prix on Sunday with Sebastian Vettel tracking eventual winner, Brawn GP's Jenson Button, for the vast majority of the race.
Button led a Brawn GP one-two, with teammate Rubens Barrichello finishing second after starting third on the grid and running strongly at the start.
Brawn's Rubens Barrichello, fighting with Vettel for second in the championship, will start fourth.
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