After Napoli were beaten by Parma on Saturday, Inter would have been confident of moving above them with a victory at a ground where they have not tasted defeat since February 2002.
However, Panagiotis Kone gave Bologna the lead with his third league goal of the season after 12 minutes, keeping his cool to steer an effort home having gone one-on-one with Samir Handanovic.
Inter had fortune on their side for the equaliser, which came six minutes into the second half.
Jonathan's strike from inside the area was heading across goal before a touch off Frederik Sorensen diverted it past Gianluca Curci and into the net.
The home side made two changes from their 2-1 defeat against Atalanta, with Cesare Natali and Diego Perez named among the starting XI, with Inter-owned duo Diego Laxalt and Jose Crespo missing out.
Inter coach Walter Mazzarri was forced into two defensive alterations due to Andrea Ranocchia's suspension and Walter Samuel's thigh injury.
Hugo Campagnaro and Juan Jesus were the men chosen to replace them, with Mazzarri also without Cristian Chivu (foot), Mauro Icardi (hernia) and Diego Milito (thigh) due to injuries.
After a low-key start, the hosts went ahead against the run of play after 12 minutes following a succession of Inter corners.
Jonathan Cristaldo carried the ball down the right wing before calmly playing in Kone, who beat the helpless Handanovic with a neat finish from inside the penalty area.
Bologna had conceded an average of two goals a game in their opening 12 matches and Rodrigo Palacio almost capitalised on their defensive frailties two minutes later when he forced Curci into a smart save with his knees.
Palacio should have done better with his header from Juan's cross after 35 minutes as Inter got into their stride and the visitors wasted a golden opportunity to level two minutes before the break.
Both Saphir Taider and Alvaro Pereira failed to turn the ball home from tight angles after Curci failed to deal with Jonathan's teasing delivery.
Bologna were on the back foot and Fredy Guarin hit the crossbar with a free-kick with the final kick of the half.
Sit came as no surprise when Inter were level six minutes into the second half courtesy of a Jonathan strike that required a decisive deflection off Sorensen's boot to beat Curci.
It was Inter's 30th league goal the season and Guarin looked set to make that 31 but he could only shoot over from inside the area after 70 minutes.
Inter piled on the pressure as they went in search of a winner and Juan struck the crossbar with a left-foot strike in stoppage time, but Bologna secured a point which leaves them a point above the relegation zone.
