It was Moreland's third career walk-off homer and his 14th home run of the season, landing in the Green Monster seats in left field. Moreland entered the game as a defensive replacement at first base to start the 10th inning.
Boston (61-49) ran its winning streak to four games and has won six of its last eight after a 9-5 win in Thursday's series opener.
Nicky Delmonico drove in the only two runs for Chicago (41-66) as the White Sox dropped their fourth straight game and lost for the 17th time in 20 games since the All-Star break.
Red Sox starter Eduardo Rodriguez threw a season-high 118 pitches in a six-inning no-decision, allowing two runs, four hits and two walks while striking out five.
Heath Hembree (2-3) picked up the win with a scoreless inning of relief. White Sox reliever Aaron Bummer (0-2) took the loss after allowing the walk-off homer.
Chicago starter Carlos Rodon lasted 7 2/3 innings in a no-decision, giving up two runs and six hits and walking none while fanning a career-high-tying 11 batters.
Boston's Craig Kimbrel found himself in a jam after Adam Engel's sacrifice bunt put runners on second and third with one out in the ninth. But Kimbrel got Yolmer Sanchez and Tim Andersen to strike out swinging to end the threat.
Nunez tied the score at 2 with his approximately 400-foot solo shot over the Monster to lead off the Red Sox's sixth.
Young ripped an RBI single with runners on first and third and nobody out in the Boston fifth to cut the Red Sox's deficit to 2-1.
Delmonico put Chicago on top 2-0 with his two-run double off the Green Monster in left field with two outs in the fourth inning.
It was the first career double for the 25-year-old Delmonico, who hit his first career home run, a three-run blast to right field, on Thursday.
NOTES: Boston manager John Farrell celebrated his 55th birthday by notching his 400th victory with the team, becoming the eighth skipper in Red Sox history to reach the milestone. ... Red Sox LHP David Price (left elbow inflammation) is experiencing general soreness and did not throw. "He's backing off it (throwing) right now," Farrell said. ... Boston 2B Dustin Pedroia (left knee inflammation) resumed light baseball activities. ... Chicago RF Avisail Garcia (sprained right thumb) took live batting practice before the game and could return from the disabled list without a rehab assignment, manager Rick Renteria said. ... Veteran RHP Al Alburquerque signed a minor league deal with the White Sox, CSN Chicago reported. ... Chicago third base coach Nick Capra had surgery to repair a herniated disc and will return next week, per the Chicago Sun-Times. ... White Sox RHP James Shields (2-3, 6.19 ERA) opposes Red Sox LHP Drew Pomeranz (10-4, 3.46) on Saturday.