The hosts got off to a flying start when coveted midfielder Toumani Diagouraga dived full length to head home David Fox’s 30-yard free-kick after 14 minutes.
Argyle surged further ahead in the 41st minute when Graham Carey followed up his saved penalty to calmly clip the ball past unlucky keeper Connor Ripley, who had made a superb stop to deny Plymouth’s playmaker from the spot.
Match referee Trevor Kettle was perfectly placed to award the penalty as Ruben Lameiras was felled on his way to goal by Bury full-back Greg Leigh.
Bury were reduced to 10 men in the 60th minute when Ryan Lowe was red carded for a foul on Argyle keeper Remi Matthews.
The home side piled on the pressure but Ripley made a string of superb saves, including a breath-taking 70th minute off-the-line aerial stop to deny Diagouraga.
Antoni Sarcevic smashed in Argyle’s third goal on 78 minutes as Plymouth made the most of their man advantage.
Gareth Bale wants first-team football and will discuss his future with Real Madrid this summer, his agent has confirmed.
Real Sociedad have firmed up their interest in Newcastle midfielder Mikel Merino by reportedly making an opening offer of £10.5m.
Freiburg defender Caglar Soyuncu says he has turned down offers from Turkish clubs in the hope a move to Arsenal will be finalised.