The star pitcher got through seven strong innings on Tuesday, leading the Giants to victory at Kauffman Stadium.
Bumgarner gave up just three hits as he struck out five in game one, putting his team on track for a third World Series title in five years.
Opposite number James Shields could only last three innings as the Giants dominated from the outset away from home, ending the Royals' eight-game winning run in the post-season.
Pablo Sandoval had an RBI double and Hunter Pence crushed a two-run homer over centre field in the first inning.
In the fourth, Michael Morse had an RBI single and Danny Duffy walked in a run as the Giants took complete control.
Joe Panik's RBI triple and Sandoval's second RBI gave the Giants a seven-run lead in the seventh.
Salvador Perez's homer to left field got the Royals on the board in the seventh, but they were out of the contest by then.
The Royals need a turnaround in fortunes on Wednesday as Jake Peavy and Yordano Ventura pitch for the Giants and Kansas City respectively.