Crawford's shot in the second inning put the Dodgers ahead 4-2, but the Braves responded with a pair of runs in the third off starter Hyun-Jin Ryu to tie it up.
Adrian Gonzalez and Skip Schumaker each delivered a run-scoring single in the next half-inning, putting LA ahead for good, and Ramirez provided an RBI triple in the fourth, sparking a four-run uprising.
Juan Uribe cracked a two-run homer and Yasiel Puig finished 3-for-5 with two RBI and three runs scored for the Dodgers, who tied the MLB record for the most runs in a post-season game while grabbing a 2-1 lead in the best-of-five National League Division Series.
Chris Capuano (1-0) earned the decision with three scoreless relief innings for Los Angeles, which will seek to close out the NLDS Monday at Dodger Stadium.
Braves starter Julio Teheran (0-1) was lit up for six runs on eight hits over just 2 2/3 innings.