The Marlins had gone 60 games without getting shut out, the longest active streak in the majors.
Atlanta, which has won four of five and 11 of 15, began the day 1½ games out of the second NL wild-card spot.
Craig Kimbrel earned his major league-high 41st save in 45 chances.
Kimbrel retired Donovan Solano on a groundout and Giancarlo Stanton on a popup before giving up a single to Casey McGehee. The All-Star closer struck out pinch-hitter Garrett Jones to end it.
Wood (10-10) matched his career high in strikeouts for the second time this month. The left-hander, who improved to 3-2 with a 1.86 ERA in his last seven starts, gave up five hits and walked none.