Julio Borbon sparked a strong start with a two-RBI triple in the second inning to give his team a 3-0 lead as the Rangers improved to 4-0 for the year.
It is the team's best start since winning the first seven games of the season in 1996.
Texas, who opened the season by crushing Boston in a three-game sweep, continued to swing the bats well with home runs from Nelson Cruz and Elvis Andrus.
Red-hot Cruz became the third player in Major League history to homer in his first four games of the season while Andrus' homer was his first since 2009.
"I was prepared in spring training and in the off-season for these moments," Cruz said. "But the main thing is winning."
Cruz's blast gave the Rangers a 4-2 advantage in the fourth and the home side stretched the lead to 6-3 with runs in the fifth and sixth innings.
Texas starting pitcher Derek Holland picked up the win after
striking out five batters and allowing three runs in six innings.
Seattle left-hander Erik Bedard allowed five runs in five innings in his first start since 2009. Berdard had undergone multiple surgeries on his throwing shoulder and was sidelined for all of 2010.
Miguel Olivo led Seattle's offense with two hits and two RBIs as the Mariners fell to 2-2. (Writing by Jahmal Corner in Los Angeles. Edited by Ian Ransom/Patrick Johnston)