The Nerazzurri only woke up during the final 15 minutes of a game that they also risked losing at the Stadio Dall'Ara, but they were unable to find a winning goal.
The result leaves them two points behind city rivals AC Milan, who thrashed Lecce 4-0 on Sunday to set the pace at the top of the table after the opening round of league matches.
Bologna dropped a bombshell on the eve of the match by sacking their coach Franco Colomba and replacing him for this game with Paolo Magnani, who was in charge of the club's youth team until less than 24 hours before kick-off.
Although Magnani is unlikely to keep the role on a permanent basis, he can be satisfied with his only match in Serie A.
His side started brightly and almost went ahead in the 14th minute when a defensive mix-up between Lucio and Julio Cesar, who were not on the same wavelength, saw the ball for Henry Gimenez, who lifted his shot over the crossbar from the edge of the area with an open goal to aim at.
Daniele Portanova should have done better with a free header from a corner in the 20th minute but he sent the ball straight at Julio Cesar from six yards out.
Gimenez missed the home side's next chance in the 41st minute when he shot wide after being found by Marco Di Vaio on the edge of the area.
Inter had lacked ideas in the first half and Benitez decided to hand Brazilian forward Coutinho his debut midway through the second half.
There was suddenly more life to the Nerazzurri's game and Wesley Sneijder forced an excellent save out of Emiliano Viviano from a direct free-kick in the 73rd minute.
The Italy international goalkeeper made a reflex save to deny Samuel Eto'o moments later as Inter raised the tempo in search of a winning goal.
Eto'o missed another excellent chance in the 76th minute when he delayed too long in a central position and Miguel Britos recovered to clear the last moment of danger for the Bologna defence.
It will be a disappointing result for Benitez, with the team having suffered a 2-0 defeat to Atletico Madrid in the UEFA Super Cup on Friday.
( ESPN.com )