The first chance fell to the home side as Francesco Bolzoni found himself well positioned, but his shot bobbled wide.
An off-target effort from Juraj Kucka was the best Genoa could manage in the opening period and the game limped to half-time.
Ten minutes into the second half Siena went ahead when Paci finished from a Francesco Valiani corner.
The game turned scrappy, with yellow cards being shown to either side but no real chances were being created.
Siena managed to hold on to their one-goal lead for a vital three points.