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.
Tottenham Hotspur joined Chelsea in round four by beating Aston Villa 2-0, but Liverpool face a replay at fourth-tier Plymouth Argyle after a second-string team were held to a 0-0 draw at Anfield.
Chelsea manager Antonio Conte made nine changes, with Terry handed a first start since October and French centre-back Kurt Zouma making his first appearance since last February following a serious knee injury.
Chelsea's 13-game winning run was ended by Tottenham 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 into the top-right corner 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 an unwanted replay if they are to join them.
Liverpool manager Jurgen Klopp made 10 changes for the visit of Plymouth, his side's average age of 21 years and 296 days making them the youngest team in the club's history.
'Whole of Plymouth'
The home side completely dominated possession, but struggled to create chances, with 17-year-old Ben Woodburn seeing a shot blocked by visiting goalkeeper Luke McCormick.
Klopp turned to his bench in the second half, sending on senior stars Daniel Sturridge, Adam Lallana and Roberto Firmino.
But although Sturridge twice fizzed shots wide and substitute Connor Smith almost put through his own goal in stoppage time, Plymouth succeeded in taking the tie to a replay.
"I don't know how many people live in Plymouth, but it felt like the whole of Plymouth was in the 18-yard box," Klopp told BT Sport.
"It is not a result we wished for, but it is not frustrating -- that's football. Plymouth deserved the draw and the rematch."
Plymouth manager Derek Adams said: "We are delighted to take them back to Plymouth."
Eight-time winners Tottenham made it six wins in a row by seeing off second-tier Villa at White Hart Lane.
Ben Davies broke the deadlock with a glancing header in the 71st minute and Son Heung-Min added a second 10 minutes from time by sweeping home Moussa Sissoko's cut-back.
Grant Leadbitter, Alvaro Negredo and Marten de Roon scored and Daniel Ayala was sent off as Middlesbrough avoided an upset by beating Sheffield Wednesday 3-0 at the Riverside Stadium.
Meanwhile, Ryan Sessegnon became the youngest goal-scorer in the full FA Cup -- not including preliminary rounds -- after scoring aged 16 years and 235 days in Fulham's 2-1 win at Cardiff City.