While the Rays do not have a losing record against left-handed starters, they dropped to 20-18 against them as opposed to 54-30 against right-handed starters.
Harrison frustrated the Rays all night, holding them scoreless on three hits while striking out eight, which eclipsed his strikeout total of seven accrued during his previous seven starts.
"I thought he pitched well; I thought we could get to him early, but once he settled in you just saw a lot of good strikes," Rays manager Joe Maddon said. "He was throwing away, then he'd get in when he wanted to. Mixed in his changeup to the righties, threw a couple of sliders to the lefties. He started pounding the strike zone. ... He was hitting the zone pretty good and I've got to give him credit for that. I thought he was very aggressive in the zone."
Harrison had walked 19 in his previous seven starts, but did not walk a batter Saturday night.
"I thought he did a good job of keeping the ball down and hitting his spots," Rays center fielder Ben Zobrist said. "He doesn't have overpowering stuff, and I know he has struggled in his last few starts. But he looked really good tonight. He looked like he was more confident tonight than he previously has been."
Already depleted by injuries to Carl Crawford and Evan Longoria, the Rays' lineup took another hit Saturday night without B.J. Upton, who was benched because he did not hustle while grounding into a double play Friday night. Upton's absence was magnified by the fact he would have been another right-handed bat to put in the lineup against Harrison.
Jackson (9-8) gave the Rays a quality start, allowing one run on five hits and three walks while striking out six in six innings.
"I liked Jack today; he was really good, and he totally gave us a chance to win that game," Maddon said.
Ian Kinsler gave the Rangers an early 1-0 lead when he homered against Jackson leading off the Rangers' half of the first.
"First-pitch fastball, he ambushed me, got a good pitch and took advantage of me," Jackson said. "Just the luck of the draw, sometimes they ambush it, but to me it's not really a mistake. He just chose to jump on the first fastball he saw."
Zobrist appeared to have tied the score when he drove a ball to deep center field in the sixth inning, but Rangers center fielder Josh Hamilton stuck his glove up at the last instant to rob Zobrist of the circuit blast.
Zobrist did not think he had a home run initially.
"I didn't think it was going to carry that well," Zobrist said. "It kept going to the wall. I got a little excited, but that got quickly taken away.
"It was a great catch, fantastic catch. He made it look so easy the way he went back on it. Phenomenal athlete. He can do so many things well. That's just another facet of his game."
Maddon felt the play made an impact.
"When Hamilton brought the ball back on Zobrist, that was a huge play, obviously," Maddon said. "So both starters I thought pitched equally well. We just could not get anything going."
Jackson gave way to J.P. Howell in the seventh and the Rays left-hander allowed an RBI double to Michael Young before Chad Bradford took over with the bases loaded and walked in another run to put the Rangers up, 3-0.
"Any loss is tough right now," Jackson said. "We're trying to get any win we can. Come out tomorrow and try to win so we can win the series. That's all we can shoot for."