The Rangers were trailing 2-1 in the seventh when two walks issued by Jim Johnson (5-2) put Yorvit Torrealba at second and Andrus at first with two outs. Josh Hamilton then blooped a hit to left, scoring Torrealba.
Reimold fielded and attempted to throw Andrus out at third, but his errant toss bounced away from third baseman Mark Reynolds, allowing Andrus to score for 3-2 Texas lead.
Adrian Beltre followed with an RBI double and the Rangers, who came into the game tied with the Los Angeles Angels atop the American League West, clinched their three-game set against the Orioles.
Tommy Hunter (1-0) earned the win in relief of Matt Harrison, who gave up 11 hits but only two runs in a six-inning start.
Mitch Atkins made his first major league start for Baltimore and held the Rangers to one run on eight hits over six frames.
Matt Wieters went 3-for-4 with a solo homer and J.J. Hardy added a bases-empty gome run for the Orioles.