Wood (1-3) earned his first victory as a Cub since coming to the team in a trade with Cincinnati. He issued four walks but used a pair of double-plays to mostly stay out of trouble.
The left-hander gave up four hits and struck out five as the Cubs clinched their three-game intracity rivalry series at U.S. Cellular Field.
Shawn Camp, James Russell and Manuel Corpas combined for two innings of scoreless relief and Carlos Marmol collected his fourth save for the Cubs -- who scored both of their runs on a two-run single by David DeJesus in the third inning.
Jake Peavy (6-3) absorbed a complete-game loss, allowing only one earned run on five hits while walking one and striking out five.
Alexei Ramirez got the only RBI for the White Sox, who fell out of first place in the American League Central.