Roma plunged further into crisis after a summer which saw Di Francesco's side lose goalkeeper Alisson to Liverpool with midfielders Radja Nainggolan and Kevin Strootman also exiting the Italian capital.
Mattiello curled in Bologna's first goal of the season after 36 minutes with Paraguayan Santander adding the second after 59 minutes.
Roma -- last year's Champions League semi-finalists -- also fell 3-0 at Real Madrid midweek, and have not won a league game since their first game against Torino.
"There's no inner fire," lamented the Roma coach. "Something is wrong."
Next weekend Roma face bitter city rivals Lazio, who saw off Genoa 4-1 on Sunday with Ciro Immobile scoring a double, to move up to fifth place.
Before the derby Roma will try to boost morale against Frosinone midweek with Lazio heading to Udinese, who won 2-0 at Chievo.
Inter Milan snatched a late 1-0 win against Sampdoria in Genoa on Saturday to move up to ninth.