Cruz went way deep, an estimated 431 feet to left-center field, to cap a five-run fifth that chased Aaron Harang (0-2).
Leonys Martin, Mitch Moreland and Adrian Beltre also homered for the Rangers, whose 11 runs were a season high. Martin had a triple leading off the fifth before scoring on Ian Kinsler's bloop single to right.
Justin Grimm (1-0) struck out nine and walked one over six innings in his fourth major-league start, the second this season. The right-hander allowed two runs, including an unearned run in the first that snapped the Mariners' 19-inning scoreless streak after they were shut out the first two games of the series.
After Kinsler's RBI single in the fifth put the Rangers up 4-2, they loaded the bases for Cruz on a fielder's choice grounder, a single and a walk.
Martin's first major league homer tied the game 1-all in the third, the first of four consecutive innings that Texas homered. That was the 19th homer of the season for Texas, the 15th solo shot, though the rest Sunday produced multiple runs.
Moreland hit a two-run homer with two outs in the fourth, followed by Cruz's slam and then Beltre's two-run homer in the sixth. It was the third homer this season for each of them, Beltre's snapping a 1-for-13 slide and Moreland's a 1-for-20.
The Rangers (12-6) overtook Oakland by a half-game for the AL West lead by completing their first series sweep since last June. Texas had lost seven consecutive series finales with a chance for a sweep since winning all three games at San Diego.
Grimm is starting in the spot of All-Star left-hander Matt Harrison, the opening-day starter who has been on the disabled list for two weeks and will have surgery Tuesday for a herniated disc in his back that will keep him out of the rotation until likely after the All-Star break.
The only other game for Grimm this season was 10 days earlier at Seattle. His scheduled start Wednesday at the Chicago Cubs was rained out, and Texas then skipped that turn in the rotation.
Michael Kirkman pitched the final three innings for his first career save.
Harang had last pitched at Rangers Ballpark 10 seasons ago, yet the results weren't any better a decade later allowing eight runs and eight hits in 4 2-3 innings. He struck out five and walked two.
In three starts at Texas for Oakland, the last on June 25, 2003, Harang had a 12.34 ERA. That figure went up to 13.22 (allowing 24 runs in 16 1-3 innings) in what was surprisingly his first loss in that stretch.
Seattle snapped its scoreless streak only two batters into Sunday's game.
Endy Chavez reached on an error leading off, getting to second base when his ground rolled off second baseman Kinsler and trickled into shallow right-center field. Kyle Seager followed with his majors-leading 10th double that stretched his hitting streak to 10 games and made it 1-0. Chavez had an RBI double in the fifth, and Seager hit a solo homer in the ninth.
Seattle hitters had 11 more strikeouts, giving them 32 in the series and 63 in five games.
NOTES: Felix Hernandez (1-2, 2.20 ERA), whose win came on opening day, is set to start Monday when the Mariners play at Houston for the first time since 2007. ... Texas opens a three-game series at the Los Angeles Angel on Monday night with Derek Holland (1-1, 1.64) on the mound. ... Seattle has never been shut out three consecutive games. ... The Rangers said fans this weekend donated tens of thousands of essential items and $40,000 in cash donations for relief efforts in West Texas, following the fertilizer plant explosion there Wednesday night. There were 80 donors during a blood drive Sunday at Rangers Ballpark.