Half-centuries to the pair as well as Jake Weatherald (59) and Jake Lehmann (57) had the Redbacks in control after a see-sawing opening day at the SCG.
Blues debutant Greg West had the visitors in trouble early at 2/51 with two wickets in three balls in the opening session, but the South Australians slowly worked their way back into the contest.
Cooper (87 not out) and Carey (66 not out) proceeded to put on a 130-run partnership as SA chase its first win of the season.
Besides Wests' opening spell, the Blues attack largely struggled on a tough SCG deck.
At one point in his maiden first-class game he had figures of 2-0, having enticed Conor McInerney to edge through to Peter Nevill and then seeing Kelvin Smith off, caught behind for a second-ball duck.
The 24-year-old was called into the side to replace Trent Copeland who is pursuing a media career with the Seven Network.