Seattle (53-50), which is 3-8 against Texas this season and 0-5 in Arlington, scored a run in the eighth inning to close within one run, but its streak of one-run wins ended at 11 straight.
"We have fed off all year on these low-scoring games set by the tempo of the starting pitching and that's what we have to do to be successful these next two months," manager Don Wakamatsu said.
Vargas was sharp over the first four innings, holding the Rangers to three hits and one run.
But in the fifth, catcher Taylor Teagarden tied the score with a one-out, solo home run to right field and, with two outs and none on, Omar Vizquel and Michael Young reached on singles and scored ahead of Byrd's home run into the left-field seats.
"It came down to one batter, I would say," Vargas said. "I had Omar 2-and-2 with two outs and had a chance to put him away. But I left a changeup up and he got on base to prolong the inning. I threw Michael Young a pretty good pitch down and away and then a changeup that cut a little to Byrd and it found the center of the plate.
"When you do that in this park, bad things can happen."
Since returning from a superb road trip to Cleveland and Detroit, winning five of seven games while holding the Indians and Tigers to two or fewer runs in four of those games, the Mariners have been rocked by home runs.
The Rangers hit five on Thursday night and two more on Friday night. Seattle has been outslugged, 21-2, in the long-ball department in the past eight games.
Something else the Mariners have been unable to do lately is get their starting pitchers deep into games, and with left-hander Jarrod Washburn being traded to the Tigers earlier on Friday, a dependable innings-eater also departed.
"We are looking for some of the younger pitchers to step up," Wakamatsu said. "[Getting deeper] into games is very critical. I think it's critical when you lose a guy like Wash, who you can almost count on seven innings every time, to save wear and tear on the bullpen."
Second baseman Jose Lopez presented Vargas with a quick lead, hitting a two-run home run in the first inning off Rangers right-hander Vicente Padilla. Lopez's 14th home run of the season also scored Russell Branyan, aboard on a two-out walk in his first at-bat after missing four games with a sore lower back.
The Rangers cut the gap in half in the second inning and went ahead in the fifth.
"I thought for the first four innings Vargas was pretty good," Wakamatsu said. "He kept the ball down, kept them off-balance. The game came down to the 1-and-2 pitch with two outs that he left up to Byrd.
"We need to get more depth from our starters and with Vargas, he made some good pitches, but on some critical ones, he couldn't finish them off, and it cost him."
Lopez had three hits and drove in three runs, shortstop Jack Wilson contributed two hits and Ken Griffey Jr. had a run-scoring single in the sixth inning, when the Mariners had three consecutive two-out singles.
"If we could have gotten even one more run there, it might have been a different outcome," Wakamatsu said.