Saunders, making his first start of the spring, allowed three hits in the first inning but managed to escape with just one run scoring.
One inning later, former D-backs outfielder Conor Jackson blasted an 0-1 offering onto the grass berm in left for a 2-0 Texas lead.
David Murphy's RBI single in the third pushed the Rangers advantage to 3-0.
Texas, which finished with 19 hits, was not done yet.
The Rangers scored another three runs in the fourth off reliever Bryan Shaw. Elvis Andrus got things going with a sacrifice fly and Nelson Cruz capped it off with a two-run single to right.
By the end of the fourth, the Rangers had out-hit the D-backs, 10-1.
Leonys Martin homered in the sixth and Yangervis Solarte's two-run homer highlighted a six-run seventh for the Rangers.
While his offense was scoring runs, Rangers starter Matt Harrison was impressive in his first spring outing. The left-hander allowed just one hit and hit a batter in tossing two innings.
David Winfree's three-run homer got the D-backs on the board in the seventh.
The win was the first of the spring for the Rangers, who had dropped their first two contests. Arizona fell to 1-3-1.
Rangers up next: Right-hander Yu Darvish will make his Cactus League debut Wednesday against the Padres and is scheduled to pitch two innings or 35 pitches at the Peoria Sports Complex. Greg Reynolds, Ben Snyder, Sean Green, Roman Mendez and Miguel De Los Santos are scheduled to follow Darvish. The game can be listened to on an exclusive free webcast on www.texasrangers.com.
D-backs up next: Trevor Cahill will make his D-backs debut Wednesday as he starts against the Indians at Salt River Fields. The right-hander was acquired by the D-backs in December from the A's in a deal that sent Arizona prospect Jarrod Parker to Oakland. Also slated to see action in the game is left-hander Tyler Skaggs, ranked by MLB.com as the team's No. 3 prospect. Skaggs was scheduled to start the D-backs' opener Saturday but was scratched due to some tenderness in his shoulder.