Seager, a career .256 hitter in four seasons, is hitting .329 against the Rangers.
Robinson Cano, Dustin Ackley and recent call-up Nick Franklin added two hits each as the Mariners, last in the American League in hitting, had every member of the batting order reach base during the first four innings,
That backed the pitching of right-hander Hisashi Iwakuma (3-0), who allowed both runs and six hits through eight innings. Iwakuma had a streak of 17 consecutive scoreless innings end with a home run by Adrian Beltre leading off the second inning that made it 1-0.
Seattle has won two consecutive games for the first time since early May.
Iwakuma began the year on the disabled list with a torn tendon in his right middle finger and made his first start on May 3. He threw a season-high 99 pitches.
The loss in the opener of a two-game set means the Rangers have either lost or split six consecutive series.
Texas' Colby Lewis (3-3) started on seven days' rest and allowed five runs on nine hits through six innings, the longest of his seven starts this season. Lewis debuted in the starting rotation in mid-April after missing the previous 21 months following elbow and hip surgeries.
Franklin was called up from Triple-A Tacoma to replace Corey Hart, who was placed on the 15-day disabled list after straining his left hamstring on Sunday. Franklin was leading the Pacific Coast League in hitting.
A Texas lineup that has yet to feature second baseman Jurickson Profar or catcher Geovany Soto this season because of injury played Tuesday without first baseman Prince Fielder and centerfielder Leonys Martin.
The Rangers hoped Fielder would return after sitting out two games because of neck stiffness that was recently attributed to a herniated disk. But after an early batting practice session, Fielder said he wasn't ready to play. Martin was removed from the original lineup because of stiff neck.
Cano has reached base in 27 consecutive games, a career high for the five-time All-Star who signed with the Mariners as a free agent following nine seasons with the New York Yankees.
NOTES: The Rangers replaced Robbie Ross in the starting rotation with Scott Baker, who will pitch Friday at Detroit. Ross, a first-year starter, was 1-4 with a 4.78 ERA . Mariners manager Lloyd McClendon said left-hander James Paxton (latissimus dorsi) and right-hander Taijuan Walker (shoulder) both threw well during game-simulated sessions on Tuesday afternoon . Texas optioned C J.P. Arencibia to Triple-A Round Rock, purchased C Chris Gimenez from Round Rock and moved LHP Martin Perez from the 15-day disabled list to the 60-day list.