Jaso's RBI single in the ninth inning capped a three-run rally off of new Texas closer Joe Nathan that pushed the Mariners past the two-time defending American League champions 4-3 on Wednesday night.
The Mariners had gone 23 consecutive innings without scoring in the series before No. 9 hitter Jaso led off the eighth with a triple into the right-field corner off Alexi Ogando and scored on Dustin Ackley's single.
"I was happy to finally get in there," Jaso said, a catcher who made 67 starts for Tampa Bay last season. "I thought it was a good debut for me in a Mariners' uniform."
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His manager and teammates would certainly agree.
Jaso was the designated hitter for his first game. Jesus Montero, the DH his first five starts, got his first action behind the plate while regular catcher Miguel Olivo got a night off.
"One of the toughest things to do when you're not playing every day is to stay ready," manager Eric Wedge said.
Justin Smoak led off the ninth with a single against Nathan (0-2) before Kyle Seager's double. Montero then had a sacrifice fly and Michael Saunders an RBI double before Jaso blooped a curveball into short center.
"The only thing that I could see is they hit a lot of sliders, not terrible sliders. I think they're looking for that pitch and waiting for it," Nathan said. "A lot of those pitches were sliders down that they served into right and found outfield turf. Credit to them putting good swings on."
Rookie left-hander Lucas Luetge (1-0) had two strikeouts in 1 1/3 innings for his first major league victory. Brandon League worked the ninth for third save in three chances.
Before the Rangers bullpen faltered, Ian Kinsler and Elvis Andrus hit back-to-back homers and Colby Lewis worked 6 2/3 scoreless innings.
Texas led 1-0 with two outs in the seventh before Kinsler hit a solo homer to left. Elvis Andrus immediately followed with an even longer shot for his first of the season.
Kinsler, who earlier in the day had finalized a new $75 million, five-year deal that kicks in next season, also doubled. But with pinch-runner Brandon Snyder at second base with one out in the ninth, Kinsler struck out before Andrus had an inning-ending liner to first.
Lewis, who had nine strikeouts in his opening day victory, struck out six with no walks against the Mariners. It was his 100th major league start in a career that began with Texas in 2002 and included two seasons in Japan before returning to the Rangers in 2010.
Alexi Ogando worked 1 1/3 innings, giving up his first run in four appearances this season when Jaso scored.
Nathan had saved his first two chances, including the Rangers' 1-0 victory Tuesday night, but lost a game Saturday night against the Chicago White Sox after giving up a leadoff homer in the ninth of a tie game.
"He'll be fine," said Michael Young, the Rangers longest-tenured player. "We have all the confidence in the world in Joe. We look forward to getting another lead and gettting him back out there."
Kevin Millwood, Seattle's oldest pitcher at 37, allowed one run with seven strikeouts and three walks over six innings in his first start of the season.
The back-to-back homers by Kinsler and Andrus came against right-hander Steve Delabar.
Earlier Wednesday, the Rangers announced the new deal for Kinsler, the two-time All-Star second baseman who twice in the past three seasons has had 30 homers and 30 stolen bases.
The 29-year-old Kinsler is in his seventh season, and his new deal includes a club option for 2018. He will make $7 million this season, and $13 million next year in the first year of the new contract.
Andrus, the 23-year-old shortstop in his fourth major league season, signed a three-year deal over the winter that takes him through his potential arbitration season.
The Rangers went ahead in the first after Josh Hamilton had a two-out double. Chone Figgins misplayed the ball, which bounced under his glove and got past him. Michael Young drove Hamilton home with a single and a 1-0 lead.
Nelson Cruz drew the second Texas walk in the first to load the bases for David Murphy, who had an AL-best five extra-base hits the first five games. Millwood started by throwing three consecutive balls to Murphy before coming back to get the inning-ending strikeout.
"He didn't have it early on. He was up," Wedge said. "It's a credit to him that he's able to find it. .... One of the best I've ever seen in regard to controlling damage. He did it in the first and later in the game."
Millwood was with Texas from 2006-09. The only active pitchers with more than Millwood's 2,565 1-3 career innings pitched or his 416 starts are 49-year-old Jamie Moyer and Livan Hernandez.
Mariners LHP Jason Vargas (1-0, 2.31 ERA) makes his third start of the season in the series finale Thursday. LHP Derek Holland pitches for the Rangers. ... Rangers C Mike Napoli didn't play a night after being hit in the helmet by a pitch. Napoli said he felt fine and manager Ron Washington said the only reason he wasn't in the lineup was because it was Yorvit Torrealba's turn to catch.