The Bianconeri lost 3-2 at the Stadio Luigi Ferraris on the final day last season but avenged that defeat, albeit in a rather scrappy encounter.
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The first half was a cagey affair, with both teams struggling to create scoring opportunities.
The scoreline remained goalless at half time but the game was livened by the opener just before the hour mark when Paul Pogba drove into the area to set up Tevez, who swept the ball past Junior Da Costa.
It stayed that way until the final whistle, although the hosts were reduced to 10 men late on as Paolo Castellini, who had only been on the pitch for a matter of minutes, received a red card for a late challenge on Stephan Lichtsteiner.