Bourn hit an 0-1 pitch into the right-field seats off Adam Ottavino (0-2). Mike Aviles singled and took second on George Kottaras' sacrifice before Bourn hit his second home run of the season.
Bourn raced around the bases and threw his helmet into the air as he was mobbed by his teammates at home plate. The walk-off win was Cleveland's fourth of the season and the first walk-off hit of Bourn's career.
Scott Atchison (2-0) worked around a two-out double in the ninth by striking out Charlie Blackmon to end the inning.
Colorado completed a 2-7 road trip and has lost four straight for the first time since last September.
Brantley's leadoff double started the Indians' third. He took third on a fly ball and scored on Chisenhall's fly out to right.
Brandon Barnes' leadoff double and a walk to Jordan Pacheco started the seventh. After a sacrifice, Blackmon's fly ball scored Barnes, who slid home ahead of Bourn's throw.
Chacin created his own problems in the first by walking three and committing a throwing error. Bourn drew a leadoff walk and took second on a wild pickoff throw. Asdrubal Cabrera walked and Brantley's single scored Bourn.
After Jason Kipnis struck out, Pacheco fielded Chisenhall's tapper in front of the plate, but his throw to second was too late to get Brantley, loading the bases.
Manager Walt Weiss went to the mound for a conversation with Chacin and the entire infield, but the talk didn't help as Giambi walked on four pitches.
Murphy's sacrifice fly scored the final run.
Dickerson's home run scored Morneau, who reached on Aviles' fielding error at third base.
Colorado's key hitters struggled during the trip. Carlos Gonzalez reached on a bunt single to start the eighth, but was 2 for 20 in nine games. Troy Tulowitzki broke an 0-for-13 streak with a third-inning single, but ended the trip in a 3-for-23 slump.
Chisenhall, normally a third baseman, made his first career start at first base.
Indians manager Terry Francona said the team's training staff thinks Carlos Santana (concussion) could return Friday. Santana has started light hitting drills and his activity will increase as he progresses. ... Rockies 3B Michael Cuddyer made his second straight start at the position. Prior to this weekend, he hadn't played there since 2010 with Minnesota. ... Indians RHP Zach McAllister (sore back) will pitch on a rehab assignment at Class A Lake County on Wednesday. ... The Rockies have Monday off before beginning a three-game home series with Arizona on Tuesday. ... Indians RHP Justin Masterson (2-4) takes on Red Sox RHP John Lackey (6-3) in the opener of a three-game series Monday in Cleveland.