Terry, making his first start since the EFL Cup exit at West Ham in October and first appearance since November, was adjudged to be last man when he fouled Lee Angol and was shown a straight red card by referee Kevin Friend.
Chelsea were already 3-0 up through goals from Pedro, Michy Batshuayi and Willian before Terry's dismissal.
Tom Nichols pulled a goal back for the League One visitors, but Pedro struck his second as Chelsea responded from their 13-game winning run ending in defeat at Tottenham last Wednesday.
Only Gary Cahill and Pedro were retained in the starting XI which began the loss at Spurs, with Terry among those hoping to press their claims for more regular action
He now faces a suspension.
Despite the changes, head coach Antonio Conte maintained his high demands and Chelsea responded with a high intensity from the start.
Terry forced Luke McGee into a sharp save when meeting a low cross from an early corner
It was the first of a number of good stops made by the Peterborough goalkeeper in the first half.
Another came soon afterwards, but McGee was powerless to stop Posh falling a goal down.
Cesc Fabregas had an effort blocked as far as the excellent Nathaniel Chalobah
McGee could only parry a Chalobah drive as far as Pedro, who paused, brought the ball back on his right foot and curled in
Pedro spurned the opportunity for a second when, six yards out, he scooped the ball on to the bar after McGee had saved from Ruben Loftus-Cheek.
Conte was frustrated when Batshuayi, desperate for a first goal in more than four months, opted to shoot himself rather than set up Willian.
Pedro cut inside to spurn another opportunity before Batshuayi benefited from a selfless team-mate
Loftus-Cheek cushioned Branislav Ivanovic's cross for Batshuayi to simply finish low into the net.
It was the Â£33million summer signing from Marseille's first goal since the EFL Cup defeat of Bristol Rovers in August.
A shock did not look likely, although two years ago in the fourth round Bradford came back from 2-0 down.
Conte put on a blue baseball cap and white cagoule for the second half due to the persistent rain.
His Chelsea side resumed full of attacking intent
Loftus-Cheek shot wide, but the Blues had their third moments later when Fabregas played a long ball wide for Pedro to chase.
Pedro cut inside again, but found Willian to finish well.
Chalobah was denied what would have been a deserved goal moments later as McGee saved well once more.
A mistake from Terry led to a foul on the edge of the area by Ivanovic on Angol
Michael Bostwick blasted the free-kick into the wall, but it was a prelude for what was to come
Kurt Zouma failed to intercept a long ball aimed at Angol, sliding on a wet pitch.
Terry was covering and went to ground in front of the Posh striker, who tried to hurdle the defender but was brought down.
Terry immediately raised his hands and turned to appeal for leniency
Friend showed him the red card.
Peterborough pulled a goal back when Angol helped on Michael Smith's cross and Nichols turned the ball in at the far post.
Chelsea had a fourth when Pedro curled in from the edge of the area, but both he and Batshuayi might have had one more each in the closing moments.
Source : PA