Adelaide were second best for much of the contest at a rain-soaked Allianz Stadium, but snatched all three points thanks to Mabil's 82nd-minute intervention.
Mabil collected a long ball down the right, before cutting inside and unleashing an unstoppable, curling left-footed drive past Vedran Janjetovic from 18 yards.
Janjetovic's opposite number Galekovic was just as crucial as Mabil in earning the win, though, as he thwarted Sydney with a string of fine late saves - the best of which came from substitute Robert Stambolziev.
The win moves Adelaide to the summit of the A-League for at least 24 hours, with Wellington Phoenix facing Melbourne Victory on Sunday.
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