The Australia native allowed three runs on five hits and a walk over 6 1/3 innings while tying a season high with five strikeouts.
"I felt really comfortable out there so I need to remember what I did," Hendriks said. "Hopefully, we can fare a bit better next time -- 11 and counting. I'm still searching for that No. 1. It'll come. I'm not too worried about it. I just need to keep throwing up quality starts and hopefully we can snag a couple."
He was again hurt by the long ball, as Alex Rios crushed a two-run shot into the second deck in the fourth inning after Paul Konerko walked. It was the 10th homer Hendriks allowed this year. He's served up a homer in each of his seven starts.
"Any time you can score a run, it's a big thing," Rios said. "Lately, we haven't been scoring so many runs like we've been doing. It's good to put the team on the board and the way Gavin pitched, that's all we needed."
The White Sox tacked on an insurance run in the seventh on a bloop RBI single from Alexei Ramirez after Rios reached on an infield single and stole second. It knocked Hendriks from the game, with Jeff Gray finishing off the inning before handing it off to left-hander Tyler Robertson, who struck out all three batters he faced in the eighth inning in his Major League debut.
"The base-hit bloop by Ramirez was a big run," Twins manager Ron Gardenhire said. "We were trying to pitch around the guy and make him chase and he was able to bloop it for a third run, which was a big one."
It proved to be an important insurance run, as the Twins rallied in the ninth against White Sox closer Addison Reed. Jamey Carroll provided a two-out, two-run single with the bases loaded, but Denard Span grounded out to second base to end the game.
"The offense came back at the end," Gardenhire said. "We had a chance there. There were some good at-bats but unfortunately, we came up just short."
The late surge from the offense came after Floyd held the Twins in check over seven scoreless frames. The right-hander scattered five hits and didn't walk a batter while striking out nine.
"Everything was working for him," said Ben Revere, who went 1-for-4 with a double. "His curveball, his slider and he was going inside and outside with his fastball and his changeup."
The Twins spoiled several early scoring chances, including in the first when Span doubled to lead off the inning but was stranded at third base with Joe Mauer striking out and Josh Willingham flying out.
Carroll also left two runners on base in the second when he grounded out to first base with runners on first and second. And Revere also doubled with one out in the third but was left there, as Mauer struck out looking before Willingham struck out swinging.
Mauer and Willingham later provided back-to-back singles with one out in the sixth, but Justin Morneau struck out and Trevor Plouffe flied out to left to end another potential rally.
It was a marked change for Floyd, who entered the game with a career 6.29 ERA in 16 career starts and eight straight losses against the Twins.
"This game is about pitching and timely hitting and he won those battles tonight," Carroll said. "I thought he had real good stuff. I've faced him a lot of times and I thought his stuff was a lot different tonight than when I faced him before."