Both sides went into the match still searching for an opening win of the campaign but it was Sassuolo who started with confidence, winning a penalty kick inside 12 minutes.
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Things then got far worse for Bologna just five minutes later, falling further behind as a deflected Antonio Floro Flores swirled into the corner
Alessandro Diamanti smashed the spot kick under the crossbar, but the visitors were unable to build momentum after that strike.
Panagiotis Kone went closest to dragging the Rossoblu level after the interval, beating two players before sending a strike crashing off the far post.
Livorno rode their luck and held on to emerge victorious from the basement battle, heaping pressure on Bologna boss Stefano Pioli as his side slip to the bottom of Serie A.