Vicente Padilla (1-0) pitched into the sixth inning to help the Rangers open 2-0 for the first time since 2000.
Ben Francisco homered for the Indians, who have been outscored 17-6 by Texas in the series.
Cruz led off the second inning with a home run off Indians starter Fausto Carmona (0-1), who allowed six runs and seven hits in five innings. Cruz added another shot off reliever Rafael Perez in the seventh that landed in the second level of seats in left.
The 20-year-old Andrus, who homered in the sixth, was in Double-A last season and he got the shortstop job before the season started. Five-time All-Star Michael Young moved to third base to accommodate Andrus.
Andrus' homer came off reliever Rafael Betancourt in the sixth.
Padilla, the Rangers' win leader with 14 in 2008, allowed four hits and three runs in 5 2-3 innings. Frank Francisco pitched a scoreless ninth for his first save.
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Coming off a 9-1 victory on Monday, the Rangers scored four runs against Carmona in the second.
Cruz hit his first homer to left-center, and Texas then had four straight hits to build the lead.
Chris Davis scored on Andrus' infield single. Ian Kinsler followed with a run-scoring double and Young made it 4-0 on a fielder's choice.
Carmona is trying to rebound after missing two months last season with a left hip strain. After winning 19 games in 2007, Carmona was 8-7 with a 5.44 ERA in 2008.
Ryan Garko was hit by a pitch with the bases loaded in the fourth to bring Cleveland within 4-1.
Josh Hamilton, who led the American League with 130 RBIs in a breakout 2008 season, drove in his first run of the season with a triple in the fifth. Cruz added an RBI groundout to score Hamilton and make it 6-1.
Francisco hit a two-run homer in the sixth that chased Padilla.
Andrus increased Texas' lead to 7-3 with his homer and he became the fourth-youngest Rangers player to hit his first big league homer behind Ivan Rodriguez, Juan Gonzalez and Sammy Sosa.
Young and Andrus each made their first error of the season.
Young's came on an errant throw to first in the fifth inning, and Andrus had a throwing error which allowed an unearned run to score in the seventh.
Texas catcher Jarrod Saltalamacchia left the game in the eighth with lightheadedness. With Asdrubal Cabrera at the plate, manager Ron Washington helped his catcher off the field, and Taylor Teagarden came into the game.
Notes:@ Indians infielder Jamey Carroll, on the disabled list because of a broken bone in his left hand, will not undergo surgery. He's expected to miss 4-6 weeks. ... Washington said SS Omar Vizquel and Teagarden should make their first starts of the season in the series finale Thursday.