Alessandro Diamanti started well and could have given Bologna a lead after seven minutes, but could not find the target with his shot.
Antonio Di Natale was starting to control the game for Udinese and he forced Agliardi into a fine save on 20 minutes.
However, despite an end-to-end finish to the half, neither side was able to find the net in the opening 45 minutes.
It took just a minute of the second half for the opening goal and it went to the hosts as Diamanti latched on to Manolo Gabbiadini's pass.
However, Antonio Di Natale scored with just over 15 minutes remaining to give Udinese a share of the spoils.