Domenico Berardi converted a penalty in the 71st minute after a handball by Spezia defender Kelvin Amian.
Berardi also hit the crossbar in the first half, while Sassuolo midfielder Abdou Harroui hit the right post late on as the home side had several chances to extend their lead.
Spezia, who surprisingly beat Inter Milan last week, remain five points above the drop zone.
Sassuolo sit comfortable in 12th, on 36 points.
AAP