The Rays continued the momentum gained from the late-season surge that earned them a wild-card playoff berth and dominated every phase of American League Division Series opener.
Texas and Tampa Bay have faced each other six times in the playoffs over the last two years and the visiting team has won every game. The Rangers will host Game 2 of the best-of-five series Saturday.
Seven of the Tampa Bay runs came on homers, two of the blasts by Kelly Shoppach, but Moore stole the show by giving up just two hits with two walks while striking out six.
Moore, 22, spent four years in the minors before making his first and thus far only regular-season start in the majors Sept. 22 against the New York Yankees. He went five innings in that game, allowing four hits and striking out five to get the victory.
He became the first pitcher to start a major league post-season contest after having no more than one start in his career.
Moore was given the surprise start for Game 1 by Manager Joe Madden because Tampa Bay's rotation had been fully extended during the final week of the season. The Rays won the wild-card berth by coming from seven runs behind against the Yankees on Wednesday for an 8-7 triumph in 12 innings.
The only two hits off Moore came from Josh Hamilton -- a two-out single in the first and a leadoff double in the fourth. Moore became the youngest American League pitcher in 40 years to start his team's playoff opener.
Johnny Damon delivered a two-run homer during a three-run second, Shoppach delivered a three-run blast in the third and then added a two-run shot in the fifth.
Tampa Bay roughed up Texas starter C.J. Wilson for six runs on seven hits in five innings.
The Rays cooled off a team that ended the regular season with six straight victories and had won 14 of its final 16 contests. Texas also led the majors in batting this year.
Tampa Bay will start ace James Shields on Saturday. In two starts against Texas over the past two months Shields allowed just one earned run in 17 innings.