Mark Noble scored from the penalty spot just before half-time to cancel out Stefan Okaka's 16th-minute strike for the Italians. Eder restored Samp's lead just after the hour-mark before Mohamed Diame levelled again for the Hammers with 13 minutes to play.
It seemed as if the spoils would be shared until Burke tapped home a cross from fellow substitute Elliot Lee to give West Ham only their second victory of pre-season.
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