Delgado (4-5) retired the first 12 men he faced before giving up Miami's two hits in the fifth -- a leadoff double by Giancarlo Stanton and an RBI, two-out single by Bryan Petersen.
Delgado also singled in what turned out to be the winning run in the fourth after Freddie Freeman had delivered an RBI base hit earlier in the inning.
Josh Johnson (3-4) suffered the hard-luck loss, allowing two runs in 7 2/3 innings with nine strikeouts.
Atlanta won its third in a row and produced its second straight outstanding pitching performance. Tim Hudson threw a five-hit shutout against the Marlins on Tuesday.