TORONTO -- Giving up a home run on the first pitch of the game didn't spoil Marco Estrada's good mood.
Shin-soo Choo homered into the center-field party deck on Estrada's first pitch, but Bautista hit a two-out drive in the fifth, his eighth home run in May after one in April.
"I didn't like that first pitch homer, obviously, but I felt really good today," Estrada said. "I woke up in a really good mood and mentally was prepared for this game."
Estrada (4-2) allowed four hits in six innings to win for the third time in four starts.
"He was great," manager John Gibbons said. "Your typical Marco start. I can't say enough good things about it."
Estrada has pitched at least six innings in 10 of 11 outings.
If he can keep doing that, we're going to stay right in this thing," infielder Ryan Goins said.
Estrada struck out eight, the fifth time this season he fanned that many. His 78 strikeouts rank third in the AL.
"He was painting his first-pitch fastball. He was painting down and away to lefties especially," Rangers outfielder Jared Hoying said. "He was hitting it all game."
Darvish (5-3) gave up three runs and five hits in six innings. He had been 4-0 with a 2.54 ERA in six starts since losing April 18 at Oakland.
Texas has lost five in a row for the first time this year, and six of seven since a 10-game winning streak.
"We've been challenged in a lot of areas," manager Jeff Banister said. "We've got to get better, obviously, we've got to put some things together."
Choo hit his 21st leadoff homer, and his first this season. It marked the eighth time Estrada has surrendered a leadoff home run and the first time it happened on the first pitch.
No. 9 batter Luke Maile walked and Devon Travis singled in the fifth, around Kevin Pillar's drive that was caught by center fielder Jared Hoying as he crashed into the outfield scoreboard. Bautista homered on a first-pitch slider.
"Jose got the hanging breaking ball and did what you should do," Gibbons said.
Hoying got the wind knocked out of him on his catch, driving his elbow into his ribs as his body slammed into the wall.
"I hit the floor, and I couldn't breathe," Hoying said.
Texas made a pair of blunders on the bases in the sixth. Nomar Mazara was safe at first when shortstop Ryan Goins dropped an infield popup, but Mazara rounded first too far and was thrown out. The inning ended when Rougned Odor stumbled out of the batter's box after hitting a slow roller past the mound, and he was thrown out easily.
MAKING THEM COUNT
Six of Bautista's home runs have tied the score or giving the Blue Jays a lead. Bautista has hit six homers with men in scoring position, matching New York's Matt Holliday for the AL lead.
Toronto is 8-2 against AL West opponents.
BLUE JAY BLUES
Texas is 8-20 against Toronto since the start of 2013 and has lost six of the past eight meetings. The Rangers have lost five straight north of the border.
Texas claimed OF-1B Peter O'Brien from Cincinnati and optioned him to Double-A Frisco. To make room on the 40-man roster, RHP Tyson Ross was moved to 60 day DL. Ross, who has not pitched this season, is eligible to be activated on June 1.
Rangers: 3B Adrian Beltre (right calf) went 3-for-7 in his third extended spring training game and could rejoin the team Monday.
Rangers: RHP Andrew Cashner (1-4, 3.18) is winless in 18 road starts since Sept. 16, 2015, the longest active road winless streak among major league starters.
Blue Jays: RHP Joe Biagini (1-2, 3.75) has allowed nine runs over his past two starts and 8 1/3 innings.