The hit completed a tour de force for Beckham, whose diving stop of Joe Panik's bases-loaded grounder generated a ninth-inning double play and likely denied the Giants a winning rally.
San Francisco fell a season-high six games behind first-place Los Angeles in the National League West while losing its fifth consecutive game and 13th out of 18. The Giants also have dropped 22 of their last 29 home games. Securing the league's second Wild Card appears to be a more realistic goal for the Giants, who trail St. Louis by a half-game for that prize.
"If we keep playing hard like we did tonight, things are going to be all right," said Giants starter Ryan Vogelsong, who worked seven innings and lapsed only when Adam Dunn hit a two-run, first-inning homer. "We have to keep showing resilience."
Giants manager Bruce Bochy expressed pride in his club's effort.
"It's a tough group," Bochy said. "They showed it tonight. I loved the way they didn't get down."
With the score tied, 2-2, in the 10th, eighth-place hitter Jordan Danks opened Chicago's go-ahead surge by drawing a four-pitch walk from Giants right-hander Santiago Casilla (1-2). Moises Sierra delivered a hit-and-run single to center field, moving Danks to third base.
With the infield playing in, Alejandro De Aza tapped a grounder to second baseman Matt Duffy, who threw out Danks at home in a play that prompted a White Sox request for a video review. The Giants' reprieve was only temporary, as Beckham grounded Casilla's 0-1 pitch solidly into left field to score Sierra.
Limited to four hits in eight innings by White Sox starter Chris Sale and trailing, 2-0, the Giants doubled their hit total while pulling even in the ninth inning.
They loaded the bases with nobody out against White Sox closer Jake Petricka on consecutive singles by Pablo Sandoval, Michael Morse and pinch-hitter Travis Ishikawa.
Then Beckham, Chicago's second baseman, executed the play of the game. He dove to his right to backhand Panik's grounder up the middle, which seemed destined to result in a two-run single. Lying flat on his tummy, Beckham reached up and flipped the ball from his glove to shortstop Alexei Ramirez, whose relay to first completed the twin killing. Sandoval scored on the play while pinch-runner Duffy advanced to third base.
Brandon Crawford, mired in a 2-for-27 skid, then singled sharply to right field on a 1-2 pitch, scoring Duffy while handing Petricka his third blown save in 11 chances.
The Giants kept applying pressure in the 10th, as Buster Posey lined a two-out single and Sandoval launched a promising-looking drive to left field. But Alejandro De Aza caught up with the ball just in front of the warning track. "Find a gap," Sandoval growled to Bochy as he passed the manager's office while leaving the clubhouse.
Chicago's eventual victory underscored the importance of Beckham's fielding gem. Bochy called it "one of the best double plays I've seen."
Said Vogelsong, " It's a game of inches, man. That ball gets by Gordon, you're talking about a whole different ballgame."
It was an entirely different game during Sale's eight-inning tenure, which featured 12 strikeouts. San Francisco advanced just two runners into scoring position against him. The first was Pence, who tripled with one out in the first inning. But he was erased when he hesitated before breaking for home plate on Posey's ensuing grounder to Ramirez and was thrown out.
"I made a mistake," said Pence, explaining that he was supposed to hold on a ball hit to third base but couldn't immediately determine the path of Posey's grounder.
Though not as spectacular as Sale, Vogelsong essentially matched him except for one critical mistake: A 91-mph, two-seam fastball which he threw to Dunn with Jose Abreu at first base and two outs in the first inning. Dunn deposited the 0-1 pitch into the left-field seats.
Vogelsong believed that the two-seamer would veer away from the plate, as it consistently did during his pregame warmup.
"For some reason this one didn't do that," Vogelsong said. "It stayed on the plate too much and he's a big, strong man. He put a good swing on it."
The White Sox lengthened their winning streak against San Francisco to six, dating back to 2008.