The Angels were 1 for 10 with runners in scoring position and left 16 runners on base, the most in a game since playing 19 innings against Boston on August 9, 2014.
Ramirez, the last of four Angels relievers, had pitched two scoreless innings before being sent out for the 12th.
Ohtani is 3-1 with a 3.58 ERA and 43 strikeouts this season. Rosario, who went 2 for 4, had singled on a ground ball past Noe Ramirez (1-2).
By the 6th inning, his team led 1-to-0. Castro was placed on the 10-day disabled list last Saturday.
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The Angels threatened in the ninth inning when Trout drew his fourth walk with two outs and took third on Upton's single to center field, but Albert Pujols hit a 105-mph line drive to center field for an out. Cozart had a pair of singles before stepping to the dish with a runner on second and one out in the bottom of the ninth inning.
Andrelton Simmons cut the margin to 3-2 in the sixth, scoring on a sacrifice fly by Luis Valbuena after getting a triple when Max Kepler couldn't make a diving catch.
ANAHEIM, Calif. (AP) - Eddie Rosario scored from first on Mitch Garver's double in the 12th inning and the Minnesota Twins beat the Los Angeles Angels 5-3 Saturday night. He gave up two runs on three hits and four walks in six innings. With his fastball hitting 99 miles per hour and his off-speed pitches plummeting through the strike zone, the Japanese right-hander reached 43 strikeouts for the season, blowing past Bo Belinsky's franchise record for strikeouts in a pitcher's first six games.
Shohei Ohtani struck out 11 while pitching three-hit ball into the seventh inning, and Zack Cozart got a walk-off RBI single in the Los Angeles Angels' 2-1 victory over the Minnesota Twins on Sunday. He struck out two and made 87 pitches.