Iwakuma (7-4) has not allowed an earned run to the Twins in five career starts, which spans 33 2-3 innings. After allowing a combined 10 runs in two consecutive starts to close out June, Iwakuma has given up just one in his two outings in July.
Fernando Rodney pitched the ninth for his American League-leading 26th save in 28 chances.
Kevin Correia (4-11) was credited for the loss despite allowing only five hits in seven innings. It was the fourth time in the last five starts that the right-hander has left the game with no run support, and the sixth time this season the Twins have not been able to score a run for him.
The Twins struggled just to put the ball in play, tying a season high by striking out 14 times.
Minnesota put runners on second and third with one out in the seventh after Kendrys Morales singled and Oswaldo Arcia doubled, half of the four hits Iwakuma allowed on the night. Iwakuma got a groundout to third and a strikeout to end the threat.
Correia faced additional trouble after Zunino's 396-foot blast to left field in the second as Dustin Ackley singled and advanced on Brad Miller's one-out walk. Correia was able to retire the next two batters, though, to avoid further damage.
Saunders' sixth home run of the year was nearly caught by Sam Fuld, but the ball tipped off the top of the center fielder's glove as he slammed into the fence.
The shutout continues a strong run by Mariners pitching, in which the staff has allowed three runs in 29 innings.
NOTES: Twins assistant general manager Rob Antony said there are no updates on 1B Joe Mauer (oblique strain) and INF Danny Santana (bone bruise in left knee), although both are progressing fine. ... Manager Lloyd McClendon said the Mariners have not decided who will start Thursday's game against the Twins. McClendon also said the scheduled starters for the weekend series against the Oakland Athletics RHP Felix Hernandez, RHP Taijuan Walker and RHP Hisashi Iwakuma is not set in stone. ... The Mariners send RHP Chris Young to the mound Tuesday to face RHP Phil Hughes of the Twins.