Australia took the lead three times, but Portugal made their better technical ability pay as they took control in the second-half.
Costa's thee goals, coupled with an impressive effort from defender Leandro Silva, secured all three points for Portugal. He almost got a late goal, which would have been his fourth of the match, but was denied by a great save from Australia keeper Lachlan Tibbles.
Australia's goals twice came from Anthony Proia, with their captain Milos Degeneck's free-kick that went in off the underside of the crossbar the highlight of their goals.
The match was the opening game of an U17 tournament that also includes England and Turkey.
Those two open their campaigns at Sincil Bank later on Wednesday. Glanford Park will also host England against Portugal on Sunday (3pm kick-off).
John Peacock, the England U17 coach, said the tournament will provide football supporters in Lincolnshire - Blundell Park is also hosting games - with a chance to see stars of the future.
"It should be a fantastic advert for youth football. The players on show this week are trying to secure their long-term future in the game [and] the experience gained from this Tournament and the Uefa qualifying tournaments can only be of a huge benefit in achieving their long-term goal," he said.