Brathwaite had defied Australia's bowlers and two rain delays before he gave away his wicket late in the day to Lyon for 85.
Brathwaite, one of the few shining lights in a dismal Windies tour with 94 in the first Hobart Test, was in sight of his fifth Test century before he fell to Lyon.
It gave the Australia off-spinner his 100th Test in Australia and 184th scalp overall.
Australia have already retained the Frank Worrell Trophy after huge wins in the first two Tests and are chasing a 3-0 series clean sweep.
Brathwaite, who along with Darren Bravo has been the best-performing batsman in the series for the struggling visitors, was deceived by Lyon's extra bounce and attempted to steer the ball past slip.
But instead he dollied a catch off his glove to Steve Smith at slip.
Brathwaite watched four batting partners come and go as Australia used two recognised spinners in Lyon and O'Keefe for the first time in a home Test for a decade.
Shai Hope, recalled for injured opener Rajendra Chandrika, began positively with some crisp drives but fell in the fifth over.
Josh Hazlewood got him to prod outside the off-stump and he was snapped up by Peter Nevill for nine.
James Pattinson dug one in and Bravo top-edged to Usman
Marlon Samuels was run out in a dreadful mid-pitch mixup before the first rain interruption in the afternoon session.
Samuels, who has scored a total of 35 runs in five innings in the series, hit towards point and darted off for a run only to stop in the middle of the pitch.
Batting partner Brathwaite had dropped his bat before retreating to his end and Samuels was stranded by Hazlewood's throw to the striker's end.
To compound Samuels' woes, rain immediately forced the players from the field and kept them off until tea was taken.
Chandrika injured his groin in the morning warm-up, after leg-spinner Devendra Bishoo hurt his shoulder on the morning of the second Melbourne Test.
The West Indies, who are playing in their first Test at the Sydney Cricket Ground in 15 years, have not won a Test in Australia since 1997.
