The 20-year-old, who is on loan at the Macron Stadium until January, tucked away a penalty just before the hour, putting his side 2-0 up after Dorian Dervite’s opener.
Bolton went 3-0 up through Jem Karacan soon after, though Jordan Rhodes’ late double reduced the deficit for Wednesday.
Armstrong played 73 minutes before being replaced by Adam Le Fondre, as Phil Parkinson’s men secured their place in the third round.
That strike was the United youngster’s second in six appearances since joining Bolton in the summer, having also netted in their Carabao Cup win at Crewe Alexandra earlier this month.