Robinson struck from the spot midway through each half to take his tally for the season to six as League One leaders Rovers maintained their unbeaten league run.
Jake Cassidy swept home Rovers' third to leave Ronnie Moore's men sitting pretty at the top of the table having dropped just four points from their opening seven games.
Bury's Efe Sodje had a 26th-minute header cleared off the line by James Wallace before Owain Fon Williams turned Stephen Schumacher's stinging drive around a post soon after.
But Tranmere took a 35th-minute lead when Danny Holmes was brought down in the box by Marcus Marshall for Robinson to score the resulting penalty.
Robinson fired in his second spot-kick of the night when Adam Lockwood felled Tranmere substitute Joe Thompson after 68 minutes. And Cassidy rounded off another impressive Rovers win just three minutes later, sliding home the rebound from Jean-Louis Akpa Akpro's shot.