Pillar, who wasn't in the starting lineup, was taking swings in the indoor cage shortly before the game when Bautista offered some encouragement.
"He just said, `Prepare yourself throughout the game, and when you get a chance, go out there and be aggressive," Pillar said. "`Have good at-bats and put yourself in a situation to help the team win."
Pillar put those words to use when he came up with runners on first and second against Casey Fien in the ninth. After falling behind 0-2, Pillar took a ball, then lifted a single into shallow right field to score pinch-runner Erik Kratz.
"It was a slider off the plate and I was just able to nestle it in somewhere," Pillar said. "And it feels good."
The first two batters of the game homered for the Twins off R.A. Dickey, but Encarnacion's 20th of the season soon put Toronto ahead.
Jose Reyes added a solo shot for the Blue Jays, who squandered a two-run lead in the top of the ninth and then recovered to avoid their first three-game skid since May 9-11. Toronto was blanked in back-to-back games Saturday and Sunday by the St. Louis Cardinals.
"The last two games have been kind of tough for us," Reyes said. "We're just happy to get this win."
Dioner Navarro drew a leadoff walk from Matt Guerrier (0-1) in the ninth and was replaced by Kratz. Anthony Gose struck out trying to bunt and Fien came on to face Reyes, who singled. Pillar followed with the winning hit and Kratz slid home headfirst ahead of the throw from Oswaldo Arcia.
"You walk the leadoff guy, you give yourself pretty much a mess and anything can happen after that," Twins manager Ron Gardenhire said.
Casey Janssen (1-0) got the win after blowing his second save.
Newly signed slugger Kendrys Morales made his Twins debut, batting sixth and starting as the designated hitter. He went 1 for 3 with a single and a walk.
The drive snapped a streak of 18 scoreless innings for the Blue Jays.
Reyes hit a leadoff shot in the fifth and the Blue Jays boosted their major league-leading home run total to 91.
Minnesota jumped out to an early lead against Dickey when Danny Santana and Brian Dozier hit back-to-back homers in the first. It was the fourth leadoff homer of the season for the Twins and Santana's first.
Minnesota tied it against Janssen in the ninth after second baseman Steve Tolleson couldn't complete a double play on Trevor Plouffe's one-out grounder to third. Pinch-runner Eduardo Nunez went to second on defensive indifference and scored when Kurt Suzuki's sinking liner dropped in front of Pillar in left for an RBI double.
"It probably wasn't the right decision, but I feel like that's a catch I've made many times before," said Pillar, who replaced Melky Cabrera to start the ninth.
Eduardo Escobar followed with a tying double into shallow left.
Dickey allowed two runs and five hits in 5 1/3 innings. He walked two and struck out three.
Nolasco gave up four runs and seven hits in 5 2/3 innings. He walked two and struck out six.
"I thought I had some pretty decent stuff tonight besides the couple of mistakes," Nolasco said. "They capitalized on that."
Morales was called out on strikes in the first, grounded out in the fourth, walked in the sixth and singled in the ninth.
Minnesota OF Aaron Hicks (right shoulder) was held out of the starting lineup but came on as a pinch-runner in the ninth. ... The Twins signed SS Nick Gordon, the fifth overall pick in last week's first-year player draft. He's the son of three-time All-Star pitcher Tom Gordon. ... Blue Jays RHP Sergio Santos (elbow) has joined Double-A New Hampshire on a rehab assignment. ... Twins RHP Kevin Correia (2-7) faces Blue Jays LHP J.A. Happ (5-2) on Tuesday.