Stoke have the worst defensive record in the division and conceded their 42nd of the season after only three minutes, when the unchallenged Rudiger headed Willian's free kick home six yards from goal.
Drinkwater doubled Chelsea's lead on nine minutes with a memorable first goal for the club, opening up his body to angle a dipping volley past a helpless Jack Butland from 25 yards.
Striker Alvaro Morata scored a hat-trick as Chelsea won the reverse fixture 4-0 in September, but he could not get the better of Butland today as the England goalkeeper twice thwarted him when one-on-one.
His compatriot Pedro had no such problems, finishing expertly into the bottom corner midway through the first half after dropping his shoulder to create the room to escape Josh Tymon from Willian's pass.
On a rare venture forward for Stoke on 39 minutes, Mame Biram Diouf thought that he had pulled a goal back with a fine finish from Charlie Adam's long-range pick out, only to be harshly penalised for handball.
Stoke's other highlight came shortly after the break, and again stemmed from an Adam pass when Saido Berahino was denied by Thibaut Courtois, who telegraphed the shot of the striker who is still searching for his first Stoke goal.
At the other end, Butland followed suit by blocking from Morata from eight yards out after Chelsea's top-scorer had been released by Willian's flick.
Willian got a deserved goal from the penalty spot on 73 minutes after he had been clumsily hacked down by Geoff Cameron, who had got himself into trouble by trying to dribble past the Brazilian in his own area.
Another poor piece of Stoke defending assisted Chelsea's fifth, although substitute Zappacosta still had plenty to do when Tymon's header fell to his left foot 22 yards out, the ball rifling into the bottom corner.
It was an emphatic way for Chelsea to sign off 2017, but Stoke head into the New Year only two points clear of the relegation zone and with defensive issues which have made them easy pickings for the top sides.