Kluber allowed two runs on five hits over seven innings, with seven strikeouts and one walk.
"It seems like every outing, he goes one more inning deeper," Indians manager Terry Francona said of Kluber. "And that's a normal progression of a good pitcher. He looks really good. He has been throwing the ball really well."
Meanwhile, Arizona's Wade Miley was not as impressive.
Miley allowed three runs on seven hits in 4 2/3 innings. His outing previous to Thursday was a regular-season start against the Dodgers in Sydney, Australia.
"I just wasn't executing pitches," Miley said. "I was leaving fastballs over the middle of the plate, really wasn't getting ahead, so I was leaving them in fastball counts so they were able to sit on things. Just wasn't very good."
The Indians got on the board in the second inning when Matt Carson doubled to right with one out, moved to third on a groundout and then scored when Nyjer Morgan legged out an infield hit.
Elliot Johnson led off the third for the Indians with a single and moved to second when Miley walked Nick Swisher.
Johnson then scored when Jason Kipnis singled up the middle as the Indians went up 2-0.
In the fourth, Morgan was credited with an infield hit when Miley was late covering first base on a grounder hit to first baseman Paul Goldschmidt.
David Murphy then ripped a double to the gap in right-center to score Morgan and give Cleveland a 3-0 advantage.
Rudy Flores and Sean Jamieson drove in runs during Arizona's two-run seventh inning after reliever Josh Collmenter worked 2 1/3 hitless innings.
"It means we've played pretty good baseball, which I love," Francona said of finishing 19-8-2 in Cactus League play. "I don't care who we're putting out there, I'd much rather win."
Up next: The D-backs will take on the Cubs on Friday night at 6:40 p.m. MST in the first of two exhibition games at Chase Field. Trevor Cahill will get the start for the D-backs. The right-hander will look to improve after struggling in a regular-season start against the Dodgers last weekend in which he walked four and hit a batter while allowing five runs over four innings. The game can be seen on MLB.TV and heard via Gameday Audio.