Sassuolo were unbeaten in eight matches at home in all competitions, but the in-form visitors took the game to the hosts from the early stages, with Dhiego Farias testing Gianluca Pegolo after a quarter of an hour.
Antonio Floro Flores, who scored for Sassuolo in the third-round win over Modena, had his side's first opportunity of the match shortly afterwards, but he could not find the target with his effort after being found just inside the box.
The game's telling moment arrived nine minutes before half time when Cagliari captain Marco Sau beat Pegolo to the ball to finish calmly and hand the Sardinians a lead to protect in the second half.
They had a chance to double that advantage before the break when Farias found Sau's striker partner Alberto Cerri, but the striker could not find a decisive connection in front of goal.
Diego Falcinelli came close to finding an equaliser for the Serie A side in the second half when his angled shot flew just wide as Sassuolo hunted for the equaliser to keep their hopes alive.