After giving up a lead-off home run to Adam Jones in the second inning, Harrison shut the lid on the Orioles, holding them hit-less through the seventh inning before giving way to the bullpen.
"After Jones hit his home run, I got a little ticked off and started making my pitches," Harrison told reporters. "Tonight, I was inconsistent with my offspeed stuff and pretty much stuck with my fastball."
He allowed two hits, a walk, and struck out three on the way to picking up the win.
Texas sent nine batters to the plate and scored six times in the third inning, adding two more in the fourth to take command.
They added another run in the seventh and four more in the ninth inning.
The Rangers offense was led by Adrian Beltre, who was 3-for-5 with a home run, three RBIs and three runs scored. Nelson Cruz had four RBIs including a three-run homer in the ninth inning, and Mike Napoli also homered.
Orioles starter Jake Arrieta lasted just 3.1 innings and allowed eight runs on six hits and two walks while striking out five.
"I am glad it only counts as one loss," Orioles manager Buck Showalter said. "I kind of dwell on Jake's (Arrieta) really good stuff, and (he) deserved to have a better outing than he did."
In the earlier game, the Orioles handed Texas their first loss of the season 5-0 behind the spectacular pitching of Zach Britton.
(Reporting by Mike Mouat in Windsor, Ontario; Editing by Alastair Himmer)