Quarter-finalists for the last two years, the Cottagers resisted the urge to rest players despite their Premier League relegation battle and duly demonstrated the true gulf in class between the top flight and League One.
Dickson Etuhu and substitutes Zoltan Gera and Jonathan Greening also found the net, while Lee Tomlin and a Grant McCann penalty salvaged some pride for the visitors, who survived an even heavier beating.
Missing playmaker George Boyd through injury, they nevertheless had their moments and even made much the better start.
David Stockdale was forced into a smart stop from Arron Davies as Fulham dithered at a corner and the goalkeeper also tipped over Craig Mackail-Smith's rising drive after a sweeping Peterborough move.
Fulham soon woke up and Simon Davies should have put them ahead against the club where he started his career when he slashed horribly at Damien Duff's low cross.
The midfielder almost made amends moments later with a sweet 20-yard volley that whistled past the post before Kamara went close from similar range and flicked a difficult header wide.
Both sides' top scorers - Mackail-Smith and Clint Dempsey - were off target before Davies curled narrowly over after some clever sleight of foot.
Tomlin blasted narrowly over from 20 yards after a lightning break from the increasingly under pressure visitors.
And they finally buckled in the 32nd minute when Chris Baird's far-post cross was nodded home with ease from point-blank range by Kamara.
Unbowed, the lively Mackail-Smith sent a dangerous ball across the Fulham box before Tomlin drew a foul from Brede Hangeland that saw the latter booked.
Duff saw a low drive deflected wide before missing a great chance to make it 2-0 after racing clear and clipping just past the post.
But the winger was the creator of the second on the stroke of half-time, his cross cleared to Etuhu, who blasted home via a deflection.
Both sides made a change during the break, Fulham withdrawing Davies for Gera and Peterborough Chris Welpdale for Nathaniel Mendez-Laing.
Posh repeated their flying start to the first half, with Mackail-Smith sending an angled drive wide and John Pantsil forced to clear acrobatically after Stockdale made a hash of a corner.
Again, Fulham got going and Gera stung the palms of England Under-21 goalkeeper Joe Lewis, Duff saw his close-range finish hit the post and screw across goal, and Kamara's attempted backheel was smothered.
Posh responded to the onslaught by handing a debut to on-loan Tottenham youngster Jonathan Obika in place of Davies in the 55th minute but they were dead and buried four minutes later.
Danny Murphy's sliderule pass broke a non-existent offside trap and Kamara raced clear before slotting the ball into the far corner.
That was the cue for Greening to replace Dempsey, but there was no respite for the visitors.
Murphy almost but Duff clean through before the latter produced an inch-perfect cross for Gera to head across Lewis and into the net for Fulham's fourth.
Everton's Kieran Agard came on for Adam Clayton in Posh's final change 20 minutes from time.
Two minutes later, Tomlin netted a consolation with arguably the goal of the game, the winger racing in from the right-hand side and arrowing a low finish into the far corner.
But there was more misery to come for Posh and more joy for Kamara when the striker sidefooted home Greening's cutback from the left to complete his hat-trick.
Hangeland was withdrawn for Rafik Halliche, who conceded a penalty for tripping Mackail-Smith moments after Stockdale had saved from Obika.
Posh captain McCann sent Stockdale the wrong way in the 86th minute but Greening made it 6-2 when he sidefooted home a minute from time and Gera also volleyed wide in stoppage-time.