The 22-year-old Baker progressed to the last 24 via a two-turn wave with six-and-a-half minutes remaining in his third-round heat to score 5.83 and finish 10.5 overall.
Only a last-ditch effort from Marco Fernandez (10.27) would deny Baker at his home break, but the Brazilian fell short of the 5.24 required.
Brazilian Deivid Silva (13.77) won the heat.
Baker will now meet fellow Australian Jack Robinson and Peru's Miguel Tudela in the fourth round, which is scheduled to get underway on Friday.
Wilson (10.03), adopted Novocastrian and world No.2, went down to Robinson (16.3) and Brazilian Caio Ibelli (12.17), in the opening heat of round three at Merewether on Thursday.
Jack Freestone was the other major casualty early in round three.
Callinan (10.76) dropped out in round two, the second heat contested on Thursday morning, despite a late surge inside the closing two minutes to avoid elimination.
The world-tour bound 26-year-old performed three turns on his final wave to land a 6.23, which left him an agonising 0.54 short of Frenchman Charly Quivront (11.3).
Brazilian Tomas Hermes (11.97) won the heat.
Round three of the women's event is due to hit the water on Thursday afternoon.