Inter's failure to beat the league's bottom side stretched their winless run to five matches, and Mancini's 10th-placed side are closer to the relegation zone than the top three.
Fredy Guarin looked to have set Inter on their way to victory with the opener after 25 minutes, but Andi Lila snatched a point for the visitors on the brink of half-time.
Mancini has overseen only five league victories since taking charge in November and, after the latest missed opportunity, he apologised for another lacklustre display.
"I am the coach and therefore I am the one who must ultimately take responsibility for this situation," he told Sky Sport Italia.
"Evidently it's my fault, as I am unable to explain properly to the team what I want from them. They are doing all they can in the circumstances.
"We had scored and started brightly, but were sluggish and didn't have the determination that is necessary at this level. We lacked grit and aggression.
"The season hadn't started well, but of course taking over is a risk. It's Inter and I couldn't turn them down. I am sorry that the situation has got worse instead of improving, but I am the coach and I take responsibility.
"There are years when things don't go right and that's the case at Inter [this year]."
