Bianchi gave a bright Bologna side the lead just before half-time, but it was a different story in the second 45 minutes.
Gonzalo Higuain and Jose Callejon turned the match around, but it was Bianchi who was to have the final say shortly after Panagiotis Kone saw red, grabbing a last-gasp equaliser.
As a result, Rafael Benitez's men fall four points adrift of second-placed Roma and 12 behind Juventus in the Serie A table while Bologna keep their heads above the drop zone thanks to the point.
Bologna, in need of points to edge away from the bottom three, took the game to the visitors, who appeared happy enough just to sit and wait for chances on the break.
The first of those, in the 17th minute, was wasted by Dries Mertens as he failed to get the ball through to the well-placed Higuain.
The Argentinian did get involved on another quick break eight minutes later, but Domenico Morleo cleared the danger with most of his team-mates still retreating from a corner.
Alessandro Diamanti led Bologna's assault, first shooting over and then sending in a cross which Bianchi sent just wide of the target.
The same two then combined for the opening goal in the 38th minute with Diamanti crossing the ball to the far post where the former Torino striker showed what he does best to beat Rafael with a header.
Napoli emerged transformed for the second half, and once Marek Hamsik took the place of Goran Pandev it was not long before they levelled.
Higuain wasted a glorious chance to score from close range before Kone fouled Blerim Dzemaili inside the penalty area and Higuain struck a venomous penalty past Gianluca Curci.
Mertens tested Curci as Napoli started to gain the ascendancy, hungry for the points after seeing both Roma and Juventus win on Saturday to pull away from them.
Bologna's resistance was finally broken 10 minutes from time when a deadly counter-attack saw Hamsik set up Higuain, who in turn teed up Callejon to put Napoli in front.
The heavens opened as the rain further dampened Bologna's day, which took a further turn for the worse with Kone receiving a second yellow card for holding back Higuain.
All was not as lost as it seemed, though, as Bianchi controlled a cross on his chest before shooting powerfully past Rafael with almost the last kick of the game in the driving rain.