Terry, handed a rare start, saw red for a last-man foul on Lee Angol at Stamford Bridge, but Chelsea progressed thanks to a Pedro Rodriguez brace and goals from Michy Batshuayi and Willian.
Chelsea manager Antonio Conte made nine changes, Terry handed a first start since October and Kurt Zouma making his first appearance since last February following a serious knee injury.
Chelsea's 13-game winning run was ended by Tottenham Hotspur on Wednesday and they were under the cosh in the early stages, with Angol and Ryan Tafazolli going close for third-tier Peterborough.
But Pedro put the hosts in front in the 18th minute, stepping inside a sliding challenge and curling home after Nathaniel Chalobah's shot was parried by Luke McGee.
After Pedro hit the bar, Batshuayi and Willian scored neatly constructed goals either side of half-time to put Chelsea in control.
Terry's 67th-minute dismissal for upending Angol gave Peterborough hope, but despite Tom Nichol reducing the arrears three minutes later, Pedro's second goal made the game safe.
Chelsea follow Manchester United, Manchester City and Arsenal in securing a place in Monday's fourth-round draw, but Liverpool must come through a replay if they are to join them as they were held to a 0-0 draw at Anfield by Plymouth Argyle.