Requiring 325 for victory, the Redbacks have withered their opposition's lead to an achievable 270 runs heading into the final day.
It was all thanks to an inspired effort in the field, with Joe Mennie (3-28) and Chadd Sayers (3-45) the pick of the bowlers.
Mennie first struck to remove double-centurion from the first innings Cameron Bancroft and the home side could never truly recover.
WA veteran Shaun Marsh scored 52 off 77 balls on Wednesday before being trapped lbw by Chadd Sayers.
Cartwright was dismissed for 18, following on from his first-innings knock of 35.
Wicketkeeper-batsman Alex Carey made an unbeaten 46, while opener Jake Weatherald scored 152 off 160 balls as the Redbacks finished their first innings on 363.
WA captain Mitch Marsh decided not to enforce the follow-on.
Warriors paceman Richardson finished with 4-90, while David Moody returned 3-95.