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Conti gave the hosts a narrow advantage heading into half time after breaking the deadlock on 43 minutes, with Cagliari's captain finding the target as a free kick deceived Daniele Padelli in the Torino goal
Giampiero Ventura's side were quick to respond after the interval, though, and levelled eight minutes into the second half through Ciro Immobile, sending a controlled effort into the far corner after fine approach play from Omar El Kaddouri.
From then on Cagliari took control of a topsy-turvy clash, with Marco Sau spurning a glorious chance to reclaim the lead as he struck straight at Padelli from close range.
Torino were unable to see out the match and fell behind again three minutes from time as Conti smashed a powerful drive into the top corner, giving Cagliari their first win in four attempts.