Los Angeles closer Brandon League blew a ninth-inning lead, but the Dodgers quickly came back to snap a two-game skid.
Crawford scored when Luis Cruz hit a sharp grounder up the middle that bounced off the foot of Betancourt. Cruz was thrown out on the play, but Crawford raced in with what turned out to be the winning run.
Juan Uribe then singled home Schumaker with an insurance tally.
League was credited with the win despite giving up a two-run homer in the ninth to Todd Helton that created a 5-5 tie. Ronald Belisario was brought on in the bottom of the 10th and picked up his first save in three opportunities this year. It was the fourth save in Belisario's six-year career.
Los Angeles let an early 5-0 lead get away, two of those runs having come in on a double by starting pitcher Clayton Kershaw in the fourth.