Colorado moved within a half game of San Francisco in a bid to escape the cellar in the National League West.
The Rockies overcame an early two-run deficit, pounded out 13 hits and made a winner out of Jhoulys Chacin -- who contributed one of the Colorado home runs.
Chacin (14-9) gave up two runs on six hits and five walks in six innings. He launched a two-run homer in the third that tied the game at 2-2. It was the first home run in his 187 career at-bats spread over five seasons.
Colorado broke the tie in the sixth with Blackmon hitting a solo shot and Dickerson adding a two-run blast. Charlie Culberson singled in the fourth run of the inning.
Troy Tulowitzki tacked on a two-run homer in the seventh.
Reliever Tony Sipp (3-2) came on in the sixth and took the loss, allowing three runs on three hits while failing to retire a batter.