Roenis Elias (1-1) worked into the seventh inning with five strikeouts for his first major league victory in the left-hander's third career start.
The Mariners went ahead to stay in the fifth on Zunino's solo homer off Colby Lewis (0-1), who made his first big league start in 21 months after elbow and hip operations.
Seattle's six-run sixth got started when Robinson Cano hit a leadoff single and scored on Michael Saunders' triple to left-center a ball that center fielder Leonys Martin bobbled without getting charged with an error. Kyle Seager followed with an RBI single that chased Lewis, and then things got interesting.
Dustin Ackley and Zunino had consecutive RBI singles after catcher J.P. Arencibia couldn't pull in a foul ball before reaching out and teetering over a camera well railing. Mix in two fielding errors, and the bases were loaded when Brad Miller hit a comebacker.
Relief pitcher Pedro Figueroa threw home, and umpire Paul Schrieber initially ruled a forceout even though Arencibia bobbled the ball on a transfer without dropping it or making a throw.
Seattle manager Lloyd McClendon challenged the call. After a review that took 3 minutes, 31 seconds, the umpires reversed their ruling, erasing the out and giving Seattle a run on the error by Arencibia.
Rangers manager Ron Washington immediately came out to argue and was ejected by crew chief Ted Barrett because managers aren't allowed to argue replay rulings. Washington's ejection came a night after Boston manager John Farrell became the first big league skipper tossed under those circumstances this season.
Umpires in New York confirmed the play was overturned because Arencibia never had secure possession while transferring the ball from his glove to his hand.
Seattle had gone 21 consecutive innings without a run before Zunino's third homer. In their previous four games, the Mariners had a 6-4 win over Oakland and scored a combined one run in the other three games.
Texas scored a single run for the third time in four games the others were 1-0 victories over Houston.
Prince Fielder, mired in a 3-for-33 slide, led off the seventh with an opposite-field double before scoring on Kevin Kouzmanoff's single. He added an infield single in the ninth.
Elias, who has a 2.16 ERA, allowed one run and five hits over 6 2-3 innings.
Lewis made his first major league start since July 18, 2012, allowing four runs (three earned) over 5 1-3 innings. He struck out four and walked one.
The 34-year-old Lewis had right hip resurfacing surgery last August. He is the first known major league pitcher to come back after the procedure to remove bone spurs without a full hip replacement.
NOTES: Logan Morrison was in Seattle's starting lineup, but didn't go to right field in the bottom of the first because of tightness in his right hamstring. Saunders replaced Morrison, who gets credit for starting the game even without batting or going onto the field. ... Blake Beavan, the Rangers' first-round draft pick in 2007, makes his 2014 debut for the Mariners on Tuesday night. Robbie Ross (0-0) pitches for Texas. Beavan was sent to Seattle as part of the Cliff Lee trade in July 2010. ... UFC welterweight champion Johny Hendricks, who trains in the Dallas-Fort Worth area, threw the ceremonial first pitch.