Pujols drove an 0-2 pitch over the center-field fence in the first inning. The three-time NL MVP increased the Angels' advantage to 6-2 in the seventh with his 12th homer this season and the 504th of his career, tying Eddie Murray for 25th place.
Price became the 319th pitcher Pujols has homered off during the regular season.
Shoemaker (2-1) was charged with a run and two hits in six-plus innings. He struck out six, five days after earning his first major league win with a 4-3 victory at Philadelphia.
Price (4-4) gave up six runs - five earned - and 11 hits in 6 2-3 innings with seven strikeouts and no walks. Kevin Kiermaier hit his first big league homer for the Rays.
It was the sixth consecutive win and 15th in 18 games for the Tigers, who have the majors' best record at 27-12.
Ian Kinsler got four hits, and Miguel Cabrera had three hits and two RBIs as Detroit dominated the first matchup between the teams since losing last season's AL championship series to the eventual World Series champion Red Sox.
It was the first time the Red Sox were swept at home by the Tigers since losing four games in June 1983.
Anibal Sanchez (1-2) came off the disabled list and went five innings for the win.
Jake Peavy (1-2) gave up five runs and 11 hits in six innings.
CLEVELAND (AP) - Yoenis Cespedes drove in a career-high five runs, Brandon Moss had three RBIs and Oakland completed a three-game sweep with a win over slumping Cleveland.
The Athletics, who have won nine of 10, outscored the Indians 30-6 in the series.
Oakland saved its best for the final game, pounding out 12 hits and drawing nine walks. Cespedes' two-run double was the key hit in a four-run fifth. He also drove in two runs with another double in the eighth.
Moss was 3 for 3 with two walks and scored four runs. Josh Donaldson, who drove in seven runs in the series, had an RBI single and scored four times.
Hernandez (5-1) continued his recent domination of the Twins, giving up two runs and seven hits while striking out five and walking one. In his past four starts against Minnesota, Hernandez has pitched 34 innings and allowed three earned runs with 30 strikeouts.
Michael Saunders tripled, doubled and drove in two runs as Seattle snapped a four-game skid.
NEW YORK (AP) - Josh Harrison homered in the seventh inning to make up for a baserunning blunder and made a diving catch in left field that helped preserve Pittsburgh's first win in the Bronx since the 1960 World Series.
Starling Marte hit a two-run homer for the Pirates in the second game after striking out four times in the opener. Marte, though, left with left hamstring tightness in the seventh. Neil Walker and Tony Sanchez homered for Pittsburgh in the opener.
Former Yankees catcher Chris Stewart had two RBIs in the second game. Fill-in closer Mark Melancon, who broke into the big leagues with New York, pitched a perfect ninth for his sixth save.
The Pirates had lost all eight games at Yankee Stadium in interleague play. They last beat New York on the road in Game 5 of the 1960 World Series.
Yangervis Solarte homered and singled for the Yankees, who had won four straight. Alfredo Aceves (0-2) took the loss.
Gerrit Cole (4-3), who turned down the Yankees when they drafted him in 2008 to go to UCLA, struck out eight in six sharp innings to help the Pirates snap a three-game skid.
In the opener, New York scored all its runs in the first two innings off Charlie Morton (0-6).
Hiroki Kuroda (3-3) allowed three runs in six innings for his first win since April 12. David Robertson got four outs for his eighth save.