The Rangers have lost eight consecutive road games, their longest drought since 2005.
Baltimore stayed within one game of first-place Toronto, which beat Milwaukee 7-4, earlier Wednesday. The Orioles (45-39) moved six games over .500, equaling a season-high.
Jones hit his 16th home run of the season and had three RBIs. Flaherty, who started at third base in place of the suspended Manny Machado, hit his fourth homer of the season.
Baltimore has homered in 12 consecutive games.
Orioles right-hander Brad Brach (2-0) entered in the sixth and earned the victory by throwing 1 1-3 scoreless innings with two strikeouts. Zach Britton earned his 11th save with a perfect ninth.