It was third time lucky for the German after mechanical failures robbed him of victory in the opening two races of the season but once he had sped past Rosberg and pole-sitter Webber to lead through the first corner, victory was never in doubt.
"It was a very good result for us especially after two races where we didn't finish where we wanted to be," Vettel told reporters after celebrating his sixth career win.
It was Red Bull's first one-two since Abu Dhabi at the end of last season.
Renault's Robert Kubica followed up last week's second place in Melbourne with another strong showing in fourth, ahead of Adrian Sutil's Force India.
The German held off McLaren's Lewis Hamilton, who was sixth, for the second half of the race.
The McLarens and Ferraris started on the back four rows of the grid after misjudging the conditions in qualifying on Saturday but Hamilton made light of starting in 20th place by surging into the points by the end of the fourth lap.
Ferrari's Felipe Massa followed Hamilton through the field to take seventh place from 21st on the grid and move into the lead in the championship after team-mate Fernando Alonso failed to finish.
Massa has 39 points, Alonso and Vettel 37 with McLaren's world champion Jenson Button on 35.
Button crossed the line in eighth place after a stirring battle with Alonso until the Spaniard retired on the penultimate lap, smoke billowing out of the rear of his Ferrari before he ground to a halt.
FLYING START
Toro Rosso's young Spaniard Jaime Alguersuari finished ninth ahead of Williams' rookie Nico Hulkenburg, the pair both earning their first points in Formula One.
Vettel got off to a flying start and stormed into the lead by the first corner and held off Webber on the second and third as the pair enjoyed an evident speed advantage to pull away from Rosberg in the first half of the race.
Webber's hopes of passing his team mate with a faster pit stop were dashed when his mechanics struggled to remove his front right wheel, allowing Vettel enough breathing space to cruise home to the chequered flag.
"The second stint was extremely long and its extremely hot here so I didn't stop sweating.. fortunately I did not run out of drinks in the car," added Vettel, who had started on pole in Bahrain and Australia.
"I was trying not to be too extreme at the beginning but it is very hot, very physical. At some stage I was hoping for rain just to get a bit of a cool down so yeah, what a day.
"Today, fortunately it stayed dry and we had a magnificent car -- the key was to pace yourself."
Further down the order, Hamilton was leading the faster cars through the field like a hot knife through butter and his charge was only halted when he ran into a stubborn Sutil, who matched the McLaren for pace on Sepang's long straights.
Button opted to pit early, as he had in his Melbourne victory, but this time he was struggling to keep Alonso behind him in the closing stages and the Ferrari's retirement on the 55th lap gave the Briton a chance to relax.
Seven times world champion Michael Schumacher continued to endure a frustrating run of results on his return to Formula One after three years in retirement, the German parking up his Mercedes on lap 10 to remain stuck on nine championship points.
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