Burnley proved to be too strong for Peterborough at Turf Moor, with Sean Dyche's side getting off to a strong start in the first half, taking a 3-0 lead inside the opening 23 minutes through strikes from Jay Rodriguez, Erik Pieters and Jeff Hendrick.
Rodriguez added a fourth goal for Burnley at the start of the second half, in between consolation goals for Peterborough from Ivan Toney and Ricky Jade-Jones, as Dyche's side earned a comfortable win to end a run of three successive defeats.
Story of the Match
Dyche had been critical of Burnley failing to start on the front during the opening half of their 2-1 defeat at home to Aston Villa on New Year's Day.
However, the Clarets, despite the eight changes made to the starting line-up, responded with a strong opening to the game against Peterborough where their crisp passing was too much for their League One opponents to defend against.
Burnley's bright start to the contest was rewarded inside the opening eight minutes, with Rodriguez, who was brought into the starting line-up in place of Ashley Barnes, tapping home the rebound after Chris Wood's header was kept out by Christy Pym in the Peterborough goal.
The Clarets swiftly doubled their advantage, with Pieters chesting the ball down and firing in a half volley from distance for his first goal for the club since his summer move from Stoke City.
Peterborough created their first meaningful chance shortly after but debutant Reece Brown could only fire wide from a narrow angle inside the area.
Burnley, though, were soon back on the front foot with Hendrick adding a third inside the opening 23 minutes with an effort from just outside the area which deflected in beyond Pym's dive.
The League One visitors continued to plug away, though, and they were gifted a potential route back into the cup tie six minutes before the break, with Matthew Lowton misplacing a header back to Joe Hart and Toney taking advantage with a well-taken effort to give Peterborough some hope.
That hope, though, was put to bed just seven minutes into the second period, with Burnley adding their fourth goal from a calm finish from Rodriguez following Aarron Lennon's intelligent reverse pass which put the forward through on goal.
Peterborough pulled another goal back 14 minutes from time, with Jade-Jones turning the ball home after Toney's effort from a corner was cleared off the line by Pieters.
Burnley could have added a fifth as the game drew to an end with Hendrick hitting the bar from a free-kick and Matej Vydra seeing his effort on the rebound deflected off target.
Burnley show the right mentality to avoid another cup upset
For Dyche one of the most pleasing aspects about the performance from Burnley against Peterborough would have been the mentality shown by his players throughout the contest, with the Clarets putting in the work needed to earn them the chance to win the game with their superior quality.
Burnley have suffered problems in the cup competitions in recent seasons when facing lower league opposition, which was demonstrated earlier in the campaign with their defeat at home to Sunderland in the League Cup.
But Dyche got the positive performance he would have been hoping for from his side.
In the past, Dyche's changes in the cup competitions did little to prove to him that they should be considered for more regular game time in the Premier League.
However, against Peterborough, the likes of Rodriguez, Hendrick, Pieters, Kevin Long and Lennon all put in strong performances to give him something to think about moving forward.
More injury worries for Dyche
The one negative thing to come out of Burnley's win against Peterborough would have been the injuries picked up by both Wood and Johan Berg Gudmundsson which forced the pair off the field, and could potentially see them miss some of the forthcoming matches.
Gudmundsson has only just come back from injury in the last few weeks, but the Iceland international suffered a hamstring injury which forced him off at half time.
Charlie Taylor came on to replace him, and that will be a real concern for Dyche given the winger is one of Burnley's main creators when he is fully fit and firing.
Wood also suffered a hamstring injury against Peterborough, but the New Zealand international's injury does not appear to be as bad as Gudmundsson's, though the striker could still now be a doubt for the next few matches for the Clarets in the Premier League.
Burnley travel to Chelsea next weekend as they look to re-find some form and momentum in the Premier League before they face another top-four side in Leicester City the following weekend at Turf Moor.