The Rangers took a 1-0 lead in the second inning on a sacrifice fly by Elvis Andrus after singles by Adrian Beltre and Mitch Moreland. They made it 2-0 in the third when Chirinos hit his fourth homer of the season.
Klein, called up Wednesday from Triple-A Round Rock, allowed four hits through four scoreless innings before giving up a leadoff homer to Xander Bogaerts, his second of the year, in the fifth.
Texas won for only the second time in six games and split the first two games of the three-game series.
Klein (1-0), who has 23 relief appearances in his career, left with runners at first and second and one out in the sixth. Shane Victorino's single off Alex Claudio loaded the bases before Brock Holt and Bogaerts were retired. It was the third time in the series that Boston left the bases loaded.
Shawn Tolleson got his first career save in 123 relief appearances after escaping a first-and-second jam in the ninth.
Joe Kelly (1-3) gave up two runs and seven hits with seven strikeouts and one walk in seven innings as the Red Sox (19-21) failed in their fourth attempt to reach .500 since they were 12-12.
Klein's first pitch of the game sailed wide and got by catcher Chirinos. But he retired 11 of Boston's first 13 batters, allowing singles to Mike Napoli in the second and Blake Swihart in the third.
With two outs in the fourth, Napoli singled and took third on a ground-rule double by Daniel Nava. Then Klein ended the threat by retiring Holt on an infield popup.
Boston threatened again in the seventh but stranded runners at first and second.
Rangers: OF Josh Hamilton went 1 for 4 Wednesday on his rehab assignment with a homer and three RBIs for Double-A Frisco in an 8-1 win over Midland. On Tuesday night, he went 3 for 3 with three RBIs for Frisco. Hamilton is not expected to join the Rangers before a three-game series starting Monday night in Cleveland.
Red Sox: 3B Pablo Sandoval missed Wednesday night's game with a sore left knee after being hit by a pitch in the seventh inning of Boston's 4-3 win Tuesday night. "He's day to day," manager John Farrell said. "We'll keep him off his feet and he'll continue to get treatment throughout the game."
Rangers: LHP Wandy Rodriguez (1-2) makes his fourth start on the road Thursday night in the finale of the three-game series. His 1.42 road ERA is the second-lowest in the AL for pitchers with at least three starts away from home. His last outing was at home Friday night when he gave up five runs in 4 2-3 innings in an 8-3 loss to Cleveland.
Red Sox: Clay Buchholz (2-4) faces the Rangers after his longest outing of the season. He allowed one run and three hits in eight innings Friday night in a 2-1 win at Seattle but was not involved in the decision. He has a 4.93 ERA despite allowing three runs or fewer in five of his eight starts.