The League One visitors took a 13th-minute lead at Gigg Lane through Lee Tomlin before Craig Mackail-Smith doubled their advantage in the 38th minute.
Ryan Lowe pulled one back in the 51st minute for Bury, who are flying high in League Two, but the hosts were unable to force a replay.
Posh were worthy of their 2-0 half-time lead after Grant McCann had an early volley turned onto the crossbar by Bury goalkeeper Owain Fon Williams before Tomlin sidefooted home from Mackail-Smith's pull-back. Mackail-Smith then headed against the bar and Bury broke forward, with Nicky Ajose beating James Wesolowski only to fire wide.
Kelvin Langmead turned a McCann cross into the side-netting but Efe Sodje's header was then cut out by Mackail-Smith, who lifted the ball over Williams for his 13th of the season.
Lowe also claimed his 13th this term, giving the hosts hope by turning in a driven cross by Mike Jones but Posh held on to progress to the third round.