Eyre (4-5) came in when rookie left-hander Kason Gabbard left the game with a stiff left forearm after throwing only one pitch in the second.
The team said Gabbard was day to day, but the injury could be concerning since Gabbard has had four arm operations in the past. The Rangers got Gabbard from Boston in the July 31 trade for closer Eric Gagne.
Byrd's single in the third made it 2-0. After Young led off the fifth with his sixth homer of the season to make it 4-0, Byrd had a double that chased Jason Hammel (1-2).
David Murphy, an outfielder who came with Gabbard from Boston, went 3-for-4 with three doubles, three runs scored and an RBI. Murphy is 4-for-6, all his hits doubles, in his two starts since the Rangers brought him up from Triple-A Oklahoma before the weekend series.
Jonny Gomes was 3-for-5 with an RBI single in the eighth for Tampa Bay. Carl Crawford was 0-for-5 with two strikeouts to end his team record of eight straight multihit games. He also had a career-best 12-game hitting streak overall, going 28-for-49 (.571) in that stretch.
Gabbard threw only 18 pitches (12 strikes) in a reunion with Florida high school battery mate Jarrod Saltalamacchia, the rookie catcher who came from Atlanta for Mark Teixeira in a trade finalized at the trading deadline like the deal with the Red Sox.
In 2000, Gabbard was a senior and Saltalamacchia a freshman starter for the Royal Palm Beach High team that made it to the 6A state tournament.
After Delmon Young led off the second with a deep fly ball, Gabbard took a moment on the mound and looked in to Saltalamacchia. Manager Ron Washington then went to the mound with trainer Jamie Reed. They only talked to Gabbard, who handed the ball to Washington and walked to the dugout.
Gabbard gave up a leadoff single to Gomes, then retired his last four batters with two strikeouts. He is 1-1 for Texas after going 4-0 in seven starts for the Red Sox.
Eyre allowed four hits and walked three, but none of them scored. The Devil Rays had the bases loaded in the fourth when Ben Zobrist lined into an inning-ending double play.
Tampa Bay loaded the bases again in the seventh -- on two walks and a hit batter -- before Frank Francisco, the fourth pitcher, struck out Delmon Young for the final out.
Murphy reached on a fielder's choice in the fourth and scored on a triple by Frank Catalanotto, who might have had an inside-the-park homer had he not jogged to first watching the ball that ricocheted off an angled wall in right-center.
After Young and Byrd got the fifth started, Brad Wilkerson had an RBI single before Jason Boots and Murphy each drove in a run with doubles to make it 7-0.
Hammel is winless in 14 major league starts, five this season. His only victory came in relief. ... Murphy is the only rookie in the majors with three doubles in a game this season. ... The game time temperature was 98 degrees, the hottest of the season. ... Rangers 2B Ian Kinsler had his 14th stolen base. He is the only player in the majors with more than seven stolen bases without being thrown out this season.