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The Rome club broke the deadlock after just 14 minutes through Osvaldo, who latched onto Erik Lamela's through-ball and fired home from the centre of the box.
Lamela himself doubled Roma's lead just two minutes later when he found the target from close range after being released by Francesco Totti.
Osvaldo made it 3-0 to the hosts four minutes before the half-time interval after combining with Alessandro Florenzi inside the Siena penalty area.
The Italian striker completed his hat-trick in the 67th minute when he shot in from the left side of the six-yard box to wrap up the three points for Roma.