Sam Winnall scored in each half for the Iron but was shown a second yellow after celebrating his second goal in the 68th minute.
Chris Sedgwick swiftly pulled one back before Nardiello, on loan from Rotherham, secured a morale-boosting point for Bury.
Scunthorpe started strongest with Dave Syers and Winnall volleying over from Deon Burton knock-downs.
Sedgwick then fired over while Bury keeper Brian Jensen turned a Terry Hawkridge shot against the post.
Bury began to put some pressure on Scunthorpe but against the run of play, Burton found Hawkridge deep on the Iron right and he squared for Winnall to tap in a 39th-minute opener.
Bury still struggled to break down Scunthorpe after the break and the visitors looked to have won it in the 68th minute, when Winnall diverted Syers' shot past Jensen.
But six minutes after being booked for simulation, the young striker was booked again for an over-zealous celebration.
Sedgwick gave Bury hope with 15 minutes to go as the winger forced in the rebound after an Ashley Grimes shot was saved.
And Nardiello latched onto Andy Procter's through-ball to sidefoot through Sam Slocombe's legs and deny Scunthorpe victory.