After recording the first two outs of the inning, Robbie Ross (4-2) walked Michael Saunders to set up the game-winning homer to right by Seager, who finished 3-for-5.
Jason Bay had a solo home run and Mike Zunino drove in a run for the Mariners, who go for the sweep on Thursday.
Charlie Furbush (2-4) threw 1 2/3 innings of no-hit ball to earn the win and Tom Wilhelmsen faced the minimum to collect his 17th save of the season.
"We didn't have any room for error with those guys coming out of the bullpen," said Seattle manager Eric Wedge. "So we needed them to pitch the way they pitched. They did a great job."
Ian Kinsler and Daniel Murphy each connected for homers for the Rangers, who have dropped three of their last four.
"Bottom line is, in the end they took advantage of the opportunities they had to win it," said Texas manager Ron Washington. "All you can do when you're facing good pitching is take advantage of the opportunities you have. We had it, we just didn't take advantage of it."
Mariners starter Felix Hernandez went seven solid innings, yielding two runs on seven hits with seven strikeouts for his third straight no-decision. Two runs of support are usually enough for Hernandez, who has a 99-25 record when Seattle puts two runs on the board for him.
Bay led off the game with a homer to left. The Rangers took the lead in the fifth thanks the homers from Murphy and Kinsler. Murphy tied the game with a blast to right-center field and two batters later, Kinsler delivered a shot to left. The Rangers went four straight games without a home run, the longest streak since June 10-14, 2011.
Raul Ibanez and Kendrys Morales each singled to start the sixth inning. After a fly out, Justin Smoak drew a walk to load the bases. Two batters later, Zunino delivered a two-out RBI single off third baseman Adrian Beltre to score Ibanez.
Derek Holland surrendered two runs on seven hits with 10 strikeouts over six frames ... The Rangers agreed to terms on a minor league contract with Manny Ramirez on Wednesday... Seattle is 16-25 on the road this season ... Leonys Martin snapped a career-high 15-game hit streak, going 0-for-4 ... Seattle went 2-for-10 with runners in scoring position, while Texas finished 0-for-6 ... The Rangers are 30-14 against the Mariners since 2009 ... Ibanez extended his hitting streak to nine games.
07/04 01:52:56 ET