After Jeon Jinwoo scored two first-half goals for the South Korean outfit on Thursday, Sydney's hopes of advancing past the quarter-finals looked to have gone up in flames.
Coach Ufuk Talay let his players know they needed to lift at halftime, and Costa led the way in front of a 7039-strong crowd at Allianz Stadium.
Sydney came out firing in the second half, with defender Alex Grant, who Talay confirmed will be named in Friday's Socceroos squad, getting the first goal to ease the Sky Blues' nerves.
Costa then turned on the magic to set up Patryk Klimala, before the Brazilian import smashed in a late penalty to put the game beyond doubt.
"I wasn't happy with the mentality and attitude in the first half, in the sense that I felt that the boys were trying to protect a 2-0 lead, rather than play our game," Talay said.
"Coming in at halftime we changed that, and we went back to the way that we play best.
"I thought we were a lot braver with the ball in the second half, which led to us capitalising."
The Sky Blues will face Singaporean outfit Lion City Sailors next month, where victory across the two legs would lead to them hosting the competition's final.
Prize money for winning the competition is close to $4 million, with Emirati side Sharjah and Saudi outfit Al Taawoun meeting in the other semi-final.
On a bouncy and uneven surface, Sydney struggled throughout the first half as Jeonbuk's Ghanaian midfielder Nana Boateng pulled all the right strings and Jeon bagged a brace.
Jeon missed a chance to seal a hat-trick early into the first half and from there Grant got Sydney back in the groove with a 59th minute goal.
Costa then took control of the game, setting up Klimala before scoring Sydney's third from the spot.
Jeonbuk coach Gus Poyet, who leaves Australia laying claim to having staged one of the great pre-match press conferences in recent memory, said he had no issues with the Allianz Stadium surface.
"I learned yesterday that there are two different rugby competitions going on here," the colourful Uruguayan shrugged.
"I wouldn't like it if I was a coach of Sydney. But again, when a pitch is bad for both teams it's not an excuse."
Poyet also revealed he was set to lose midfielder Jeon Byung-kwan, who has been conscripted to serve in the military, this month.
