Destro's 12 goals were crucial as Siena avoided relegation last season but the Tuscan outfit were unable to hold onto his services in the close season as Roma landed their man following protracted negotiations.
Siena looked poised for a measure of revenge against the capital club when Neto headed them in front on Sunday, but Destro struck either side of a Simone Perrotta goal as Roma hit back for all the points after the break.
The hosts had much the better of the opening period, with Simone Vergassola blazing over and Emanuele Calaio firing just wide inside the opening 23 minutes.
The breakthrough came in the 26th minute as Francesco Valiani's corner found its way to Neto who rose highest to head powerfully into the top corner.
The goal stung the visitors and Francesco Totti twice went close to levelling before the interval, the forward first seeing his drive saved by Gianluca Pegolo and then hitting a rising shot over the bar.
Roma's pressure finally told in the second period as Italy Under-21 star Destro took centre stage for the second week running.
The 21-year-old scored the only goal as Roma saw off Pescara last weekend, and he opening his account on Sunday in the 63rd minute with a close-range header from Alessandro Florenzi's lofted cross.
The result remained in the balance until four minutes from the end when Perrotta capitalised on a loose touch by Siena's Matteo Rubin to lash home low from the edge of the area.
And the game was finally put beyond Siena in stoppage time when Destro steered in Miralem Pjanic's inviting cross with a controlled left-footed finish.