Hamilton had a two-run drive in the first inning, taking the major league lead and matching his career high. He added an RBI single in the third to give him 99 RBIs, which is also tops in baseball.
Darvish (12-8) allowed six hits and walked five in 6 2-3 innings. He is 3-0 against the Tigers this year.
Ian Kinsler drove in two runs for the AL West-leading Rangers. Miguel Cabrera had a two-run double for the Tigers.
Rick Porcello (9-7) gave up nine hits and six runs.
BLUE JAYS 10, YANKEES 7
TORONTO (AP) — Rajai Davis doubled twice, matched a career high with five RBIs and made a sparkling catch as the Toronto Blue Jays beat the Yankees and snapped a five-game losing streak.
Davis scaled the left-field wall in the seventh to snare a potential long ball off the bat of Casey McGehee. Davis also stole his 33rd base.
Edwin Encarnacion hit his 30th homer for the Blue Jays.
J.A. Happ (1-1) pitched 5 2-3 innings for his first win since joining Toronto in a July 20 trade with Houston. Casey Janssen got his 15th save in 16 chances.
Robinson Cano and Derek Jeter homered for the Yankees, who lost for the first time in five games.
Phil Hughes (11-10) took the loss.
MARINERS 4, ANGELS 1
Jesus Montero homered twice off major league ERA leader Jered Weaver, and Jason Vargas outpitched his former Long Beach State teammate to lift the Seattle Mariners over the Angels.
Weaver (15-2) had won nine straight outings. He fell one shy of Chuck Finley's 1997 franchise record for consecutive starts won.
Vargas (13-8) allowed a run and seven hits over 8 1-3 innings. The left-hander gave up a one-out triple in the ninth by Howie Kendrick before Tom Wilhelmsen came in and got the final two outs for his 16th save in 18 chances.
Weaver was 6-0 with a 2.23 ERA and 26 strikeouts in six starts during July, but Vargas beat him out for AL pitcher of the month honours with a 5-0 record, a 1.64 ERA and 26 strikeouts in his six starts.
ORIOLES 5, ROYALS 3
BALTIMORE (AP) — Manny Machado hit his third homer in four major league games, Nick Markakis also connected, and the Baltimore Orioles got four hitless innings from their bullpen in a victory over Kansas City.
Machado hit a two-run homer in the second off former Orioles pitcher Bruce Chen (8-10). Machado tripled in his debut Thursday night, homered twice in his second game and doubled in a run Saturday night.
The 20-year-old Machado joined Frank Robinson, Ray Knight and Lee May as the only players to get an extra-base hit in each of their first four games with the Orioles.
Mike Moustakas homered for the Royals, who went 4-3 on a road swing against the White Sox and Orioles.
Luis Ayala (4-3) worked the sixth, and Troy Patton and Pedro Strop each got three outs before Jim Johnson finished for his 34th save in 37 opportunities.
RED SOX 14, INDIANS 1
CLEVELAND (AP) — Jon Lester struck out 12 over six innings for his first win in six weeks and the Boston Red Sox salvaged a four-game split by routing Cleveland.
Adrian Gonzalez hit a two-run homer and two-run double. Carl Crawford had three doubles and threee RBIs.
The Red Sox built a 6-1 lead against Corey Kluber (0-1) and scored their most runs since touncing Miami 15-5 on June 20.
Lester (6-10) was on cruise control with the big lead after struggling to an 0-5 record and 7.49 ERA in seven starts since beating Toronto on June 27. He gave up three hits and one run to move to 5-1 lifetime against Cleveland.
Boston won for only the fourth time in 12 games.
RAYS 7, TWINS 3
MINNEAPOLIS (AP) — Jeff Keppinger homered and doubled, then drove in the go-ahead run in the 10th inning as the Tampa Bay Rays beat Minnesota for their sixth straight win.
Ryan Roberts had three hits, including a two-run single in the 10th. Pinch hitter Matt Joyce capped the burst with an RBI double as Tampa Bay matched its longest winning streak of the season.
Kyle Farnsworth (1-3) worked around a one-out single in the ninth to earn his first win since last Aug. 21. Fernando Rodney retired the only batter he faced for his league-leading 36th save in 37 attempts.
Alex Burnett (4-4) took the loss in Minnesota's fifth defeat in a row.
WHITE SOX 7, ATHLETICS 3
CHICAGO (AP) — Chris Sale struck out 11 in 6 2-3 innings, A.J. Pierzynski had a two-run home run in a five-run sixth inning and the Chicago White Sox beat Oakland.
Sale (14-3) reached double digits in strikeouts for the third time this season. He fanned 10 in the first five innings, allowed two runs on solo homer and walked none.
Bartolo Colon (9-9) allowed six runs and nine hits in 5 2-3 innings.
With the score tied 1-all in the sixth, Pierzynski's 23rd homer put the White Sox up 3-1.