Zaza got ahead of a tired-looking Dutch captain Virgil van Dijk to steer the ball home from close range.
But Italy had defender Domenico Criscito sent off two minutes later after he picked up a second caution, opening the door for the Dutch to push for an equaliser.
Substitute Ake stole in the back post unmarked to power home a header for the 88th-minute equaliser and might have grabbed a winner on the stroke of full-time in similar fashion.
Italy should have been comfortably ahead at halftime even though both sides looked leg-weary.
Dutch keeper Jasper Cillessen made a point-blank save from Andrea Belotti and Simone Verdi missed a sitter for Italy.
Dutch midfielder Rudi Vormer cleared off the line with a header just before the interval to deny Bryan Cristante.
Both teams are rebuilding under new coaches -- Roberto Mancini for Italy and Ronald Koeman for the Netherlands.
(Reporting by Mark Gleeson in Cape Town; Editing by Ed Osmond)
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