Brathwaite hit an unbeaten half-century and with Bravo steadied the tourists' innings in an unbroken 79-run partnership following the early dismissal of Shai Hope.
At lunch, the West Indies were 92 for one with Brathwaite on 50 and Bravo not out 32.
The pair have been the two best-performing batsmen in the series for the struggling Windies, with Bravo scoring 108 in Hobart and 81 in Melbourne, while Brathwaite hit 94 in the first Test.
Josh Hazlewood got him to prod outside the off-stump and he was snapped up by Peter Nevill for nine.
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Chandrika injured his groin in the morning warm-up following on from leg-spinner Devendra Bishoo injuring his shoulder on the morning of the second Melbourne Test.
Spinner Nathan Lyon was introduced into the Australian attack in the eighth over, indicating that spin is likely to make a significant impact over the course of the Test.
Fellow spinner Steve O'Keefe was brought on for his first bowl in the 27th over to applause from his home SCG crowd.
The West Indies, who are playing in their first Test at the Sydney Cricket Ground in 15 years, are chasing their first Test win in Australia since 1997.
The SCG floodlights were switched on at the start of play in gloomy conditions with the threat of rain later in the day.