The Shakers trailed 2-0 at the break and looked out for the count.
But they were handed a dubious penalty for a mystery infringement by Marcus Williams and Thomas Kennedy duly converted the spot-kick.
The goal breathed new life into the League Two side and right-back Paul Scott looped in a 70th minute equaliser to send the teams back to Glanford Park on November 15.
However, it needed a fine save from Craig Dootson in the third minute of stoppage time to deny Andy Keogh the winner.
Keogh, who scored against Scunthorpe last season while at loan at Gigg Lane, headed in past Dootson after only five minutes.
Bury squandered several chances to get back on level terms before Ian Baraclough hit a spectacular volley to double Scunthorpe's advantage after 41 minutes.
Keeper Paul Musselwhite was so incensed by the penalty award, given for a foul on Simon Whaley, that he was booked for protesting after the final whistle.
But it wasn't Scunthorpe's day as Peter Till injured his ankle in the warm-up and had to be replaced five minutes before kick-off by Matthew Sparrow.
But Mr Kettle's decision baffled virtually everyone in the ground and left Scunthorpe boss Brian Laws to say: "Tell me why that was a penalty.
"I don't think anyone could see why he gave it. It gave Bury a lift and they threw caution to the wind after that and got a second goal.
"But in the end we could even have lost that game so the most important thing is to still be in the competition."
Bury boss Chris Casper, after his first ever FA Cup tie as a manager, said: "We had a bit of luck with the penalty but overall I think we deserved it. We are not going there just to make up the numbers.
"I've told the players that they are idiots if they don't believe they are a match for anyone in our division and even in the next league."