Moncada hit a go-ahead double in the 10th inning, lifting the White Sox over the Tampa Bay Rays 3-2 on Friday night.
Nicky Delmonico had a two-out double in the 10th off Jose Alvarado (1-4) and was replaced by pinch-runner Adam Engel. Tim Anderson was intentionally walked, and Moncada hit a liner over left fielder Jake Bauers.
"This is my chance," Moncada said through a translator. "I have to take advantage of it, Just try to make a good swing, make contact and hit it."
Tyler Danish (1-0) struck out Jesus Sucre with two on and two outs in the ninth, and Hector Santiago got his first save since 2012. He walked Mallex Smith leading off the bottom half of the 10th, Kevin Kiermaier popped up a bunt and Matt Duffy grounded into a double play.
Danish, from nearby Brandon, Florida, made his first appearance at Tropicana Field.
"I know there was a couple people up there screaming, so it was exciting," Danish said.
Jose Abreu homered for the White Sox, who won for the ninth time in 24 games and are just 39-70. Tampa Bay is 20-5 at home since June 11.
Giolito allowed one run and two hits in seven-plus innings, holding Tampa Bay hitless until Kiermaier singled starting the fourth.
"I think the story was Lucas keeping us in the game the whole game," White Sox manager Rick Renteria said. "He got to 99 pitches. He gave us a great outing."
Giolito said he might have thrown "like five strikes" during his pregame bullpen session.
"But it was one of those days where I got warmed up and I said `All right, let's see what happens," Giolito recalled.
Abreu's 18th homer, a two-run drive in the sixth off Jaime Schultz, gave the White Sox a 2-0 lead. Abreu has eight homers and 28 RBI in 28 games against Tampa Bay.
David Robertson doubled leading off the eighth for the Rays' second hit and scored when Joey Wendle singled off Jace Fry. Wendle stole second, advanced on a wild pitch and scored on Mallex Smith's bunt single against Juan Minaya.
"It's frustrating," Bauers said. "Another good battle back. We just didn't quite finish it out this time."
The Rays top the majors in one-run games (44) and single-run losses (24).
Robertson sprained his left thumb on his slide at second and was replaced at shortstop in the ninth by Willy Adames.
Tampa Bay manager Kevin Cash said it's likely Robertson will be going on the disabled list.
"Wasn't able to make a grip," Robertson said.
Tampa Bay is honoring its 2008 AL championship team this weekend. The Rays lost their first -- and thus far only -- World Series in five games to Philadelphia. 2B Aki Iwamura kept the ball from the forceout he recorded in the AL Championship Series Game 7 win over Boston. "That's the year that changed my life and career," Iwamura said through a translator. Cash was with the Red Sox in 2008. "They shocked baseball," Cash said.
Rays: OF Tommy Pham (broken right foot) went on the 10-day DL. He hopes to be back in two to four weeks.
Rays LHP Blake Snell (12-5), out since the All-Star Game on July 17 with left shoulder fatigue, returns Saturday night to face the White Sox and LHP Carlos Rodon (3-3). Tampa Bay optioned RHP Austin Pruitt to Triple-A Durham to make room on the roster for Snell.
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