A foot injury meant Brook had been in the ring just once since beating Jones for the first time a year ago, and also forced the 27-year-old to withdraw from a IBF welterweight title showdown with Devon Alexander in April.
Brook survived several scares against Carson last July but it was more comfortable on a steamy night in Hull on Saturday night with the American stopped in the eighth round after a barrage of landing blows.
World super-middleweight champion Froch was impressed with Brook's performance and told Fight Night Live: "He got a fantastic start and we were all jumping out of our seats and expecting him to close the show.
"But he took his time and that shows he has learned. It was a great performance. We feel it could have ended sooner but he did what he needed to do and he finished in style."
It was a contrast to their first encounter in Sheffield a year ago when Jones overcame a sloppy start to bloody Brook's nose in the eighth and pushed viciously for a knock-out before losing to the home fighter on points in the IBF world title eliminator.
"Brook would have wanted to draw a line under that fight," Froch added. "That first fight with Jones was a nightmare and now he's got the stoppage and so he can draw a line under the whole saga.
"He's been messed around with the Devon Alexander fight - not that it's Alexander's fault because he's been injured and so has Brook.
"But it's now a new chapter and he can move forward with Carson Jones out of the way."