ARLINGTON, Texas -- The Texas Rangers patiently awaited their opportunity to get to Baltimore ace Chris Tillman on Thursday night.

That persistence was rewarded.

The Rangers scored five times off Tillman in the sixth inning to rally past Baltimore 6-3 at Globe Life Park. Texas moved over .500 at 6-5 while handing the Orioles (7-2) their first losing streak of the season after a 7-0 start.

Texas was lifeless offensively heading into the bottom of the sixth, trailing 3-1 and being four-hit by Tillman.

However, the Rangers sent nine batters to the plate in the sixth, stringing together four consecutive run-scoring hits to grab a 6-3 lead.

"(Tillman) was making great pitches," said Texas second baseman Rougned Odor, whose two-run double snapped a 3-3 tie. "He was throwing a lot of sinkers, curveballs and changeups. We figured it out, and we just hit it good.

"We have a pretty good offense. We can drive in runs. As soon as we score one run, then comes more."

Odor's liner to left was the hardest-hit ball of the inning. Texas got its scoring started in the sixth on a bloop double from Mitch Moreland. Ian Desmond's infield single tied the game before Odor put the home team in front with his double.

Elvis Andrus lined a single up the middle for the final run of the frame.

"We have a mindset to capitalize on those types of innings, how we come back when we're down, how to stabilize when we have the lead and really staying on the ball, staying through the ball and knowing that singles and doubles are good also," Texas manager Jeff Banister said.

Tillman (1-1) would up allowing six runs on nine hits in his 5 1/3 innings.

"That kind of stuff happens, and you've got to continue making pitches," Tillman said of the sixth inning. "That's our job as starters. No matter what happens, you've got to go out and make another pitch and make another pitch. I left some balls out over the plate that they hit hard, too. I feel like I made some decent pitches throughout. I feel like I had the stuff to get deep in that game."

The Rangers' rally took lefty Cole Hamels off the hook for his first loss since last Aug. 7 and made a winner out of reliever Tom Wilhelmsen (1-0). Hamels allowed three runs on seven hits in 5 1/3 innings, and Texas won its 13th consecutive game in which he started.

Hamels wasn't sharp, but he made the pitches when he needed to as Baltimore left seven on base and was 2-for-7 with runners in scoring position.

Baltimore manager Buck Showalter knew the Orioles had to take advantage of their chances against Hamels.

"I think he's won a Cy Young because they bend but they don't break," Showalter said. "That's why those guys are in such demand and the Rangers wanted him. That's what they do."

Wilhemsen faced one batter and induced an inning-ending double play in the sixth. Keone Kela and Sam Dyson followed with a scoreless inning apiece, and Shawn Tolleson worked around a two-out single in the ninth to pick up his third save.

The Orioles jumped on Hamels for a run in the third on back-to-back doubles, with Joey Rickard's liner to left putting Baltimore on top 1-0.

The Rangers came back in the bottom of the frame to tie the game against Tillman. Mazara flared a single to center to score Andrus, who opened the frame with a single.

Mark Trumbo's solo homer with two outs in the fourth snapped the 1-1 tie. J.J. Hardy's sixth-inning single put Baltimore on top 3-1.

NOTES: The Rangers are in the midst of a 14-game stretch to open the season, the only team in baseball that is opening the season with that many consecutive games. It is just the third time the Rangers have opened a season with at least 14 straight games, with the other two coming in 2005 (20 games) and 1992 (14 games). ... Texas starters opened the season by going at least five innings in 10 straight starts for just the fifth time in club history. The last time it happened was 2012. ... Baltimore's seven-game winning streak was its longest under manager Buck Showalter. The last time a Showalter team won more than seven straight games was in May 2005, when he managed the Rangers. ... Baltimore CF Adam Jones, who had been out of the starting lineup since April 6 due to a strained rib cage, batted third and went 0-for-4. ... Rangers RHP Yu Darvish said he felt fine after throwing 30 pitches of live batting practice Wednesday in Arlington. Darvish, who missed last season because of elbow surgery, could make his first rehab start later this month.