"It means a lot," said Holland, who missed the first five months following offseason knee surgery. "Especially to get Bogie's first win."
Texas gave Bogar, its interim manager, his first victory in the third game since Ron Washington resigned.
"The (game) ball will end up somewhere," Bogar said, "and the lineup card."
Seattle, which had won five in a row, finished a 5-2 road trip.
Leonys Martin had three hits, including a triple, and scored the only run. He seemed energized after hearing that Bogar intends to keep him in the lineup every day.
"I love the communication," Martin said.
He led off the sixth with his second single and stole second base. Daniel Robertson's groundout sent Martin to third, and he scored easily on Beltre's fly to deep center field.
Holland (1-0) allowed six hits and no walks on 93 pitches while striking out five.
"I feel like I could go longer," Holland said after his second straight seven-inning start. "I was efficient at first and did a good job keeping my pitch count down until they had some long at-bats. I kept the defense on their toes."
The left-hander pitched out of trouble in the seventh. Robinson Cano led off with an infield single, his second hit. With two outs, Cano advanced on a passed ball and went to third on Corey Hart's infield hit. Holland then retired rookie Stefen Romero on a grounder.
"That's what good pitchers do -- they shut them down. He's a proven pitcher. He threw a hell of a game," Seattle manager Lloyd McClendon said.
Hart's hit was the only one with a runner in scoring position for either team. Seattle was 1 for 9 in those situations. Texas went 0 for 7, extending an unproductive streak to 0 for 19.
Mariners: McClendon said OFs Dustin Ackley (ankle) and Michael Saunders (strained left oblique) could return Monday. Ackley sat out after leaving Saturday's game. Saunders has been on the disabled list since July 11.
Rangers: C Robinson Chirinos (stiff neck) missed his seventh straight game. "I hope it turns around pretty quick. We could use him," Bogar said.
The Mariners return home to open a three-game series Monday against Houston. RHP Felix Hernandez (14-5, 2.18 ERA) is scheduled to face the Astros. He leads the AL with 24 quality starts and is second in ERA.
Texas is off Monday before opening a three-game series Tuesday against the Angels. RHP Colby Lewis (9-12, 5.42) is the scheduled starter.
The shutout was Texas' 17th this season. The Mariners were blanked for the 16th time. Both rank second in the American League to Tampa Bay, which has pitched 19 shutouts and been held scoreless 17 times.
The Rangers tied a team record with their fifth 1-0 victory this season.
WHO'S NO. 3?
"I'm trying to find a way to get Beltre pitches. I don't know if that will ever be a possibility this year," Bogar said.