Nathan Stanton was a tower of strength for the Iron but strikers Andy Keogh and Billy Sharp were given little room to work their magic.
Bury's Simon Whaley was a key player, receiving sound support from Dave Flitcroft and Dave Challinor and they were never out of the hunt.
However, Bury's efforts ended with more problems when Flitcroft was shown a straight red card at the end of the game for what appeared to be comments to the referee.
Teenage striker Sharp was first to show for Scunthorpe, making a brilliant 40-yard run before driving a clear shot over the bar.
Bury responded immediately with Whaley watching a long-range free-kick whistle inches wide and he followed up with another excellent effort which struck keeper Paul Musselwhite on the chest before he could gather.
On the half-hour mark Keogh made a brilliant run into the heart of the Bury defence before crossing for Sharp who half-volleyed a superb shot over the bar.
Shortly afterwards Neil Edwards for the visitors did well to beat out a cracking effort from Ian Baraclough.
Defences were almost in total command in the second half but the visitors almost snatched a goal in the 70th minute when Whaley broke free to drive wide when in a good position.
With two minutes of normal time remaining Scunthorpe had a late chance when Keogh blazed wide.
Extra time saw Andy Butler lift a good chance over and substitute Matt Tipton was inches wide for the visitors.
However, the game ended quite dramatically with Scunthorpe snatching a trip to Aldershot when Johnson powered home the winner and Flitcroft saw red.