Barrow scored his fifth goal in the last six games to earn his side a deserved point in a game in which they also had two goals ruled out by the Video Assistant Referee for marginal offside decisions.
Kostas Manolas netted an early goal for Napoli, who move to 53 points from 33 games, realistically out of the running for the Champions League qualification places but having already secured a Europa League place via their Italian Cup win last month.
Bologna's poor run at home continues as they stretched their winless streak to six matches, a major contributor to their mid-table position, but they deserved more from the contest.
The visitors were ahead inside seven minutes thanks to a diving header from Manolas, as he launched himself at Matteo Politano's corner.
Bologna thought they had found an equaliser when defender Ibrahima Mbaye headed the ball into the net following another corner, but the goal was ruled out for offside.
Napoli winger Hirving Lozano dribbled past four defenders in a mesmerising run before firing narrowly wide in the second half, while at the other end there was more frustration for the home side as they had a second goal ruled offside, this time from Rodrigo Palacio.
They finally got their reward when Barrow found space on the edge of the box and applied a neat finish to keep his goalscoring run going.
There was more bad luck for the home side when Danilo's thunderous shot late on struck the woodwork.