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After Adelaide broke the deadlock in the first half, the Glory turned in a strong second-half performance, netting the winner in the 83rd minute when substitute Daniel Stynes' shot took a deflection which wrong-footed goalkeeper James Delianov.
Perth coach Richie Garcia praised his side's determination to respond to their first half setback.
"I definitely think in the second half we got to grips with what we were doing," he said.
"The character and the willingness to get in front and get back in the game was notable from our players."
Perth looked the livelier of the two sides in the opening quarter hour with Andy Keogh, Bruno Fornaroli and Diego Castro finding plenty of the ball and space and causing the Reds defence concern.