A rainout Saturday night forced Sunday's doubleheader.
De La Rosa got his ninth win in 10 decisions as the Rockies remained 1½ games ahead of San Francisco in the NL wild-card race. Florida fell two games back.
"Every win is very important for us now, especially over this team," De La Rosa said. "They are fighting with us for the wild card."
Barmes' two-run homer off Rick VandenHurk gave the Rockies a 3-0 lead in the second. Barmes drove VandenHurk's pitch over the American League scoreboard in left for his 18th home run of the season.
"Every win we can get is a big win," Barmes said. "There is no game we can give away at this point."
The Rockies increased their lead to 4-0 on Carlos Gonzalez's second homer of the doubleheader and sixth of the season in the third. Gonzalez also homered in the first inning of the first game.
"This is a very good baseball club that we just faced," Rockies manager Jim Tracy said. "Their approach, to their credit, was relentless. We had to work really hard in the second game to win the game."
De La Rosa (11-8) left after two outs in the sixth. He allowed seven hits and one unearned run, walked three and struck out nine.
"I felt a little bit tired in the fourth," De La Rosa said. "But I made good pitches when I was in trouble. That helped me."
Hawpe's two-out, two-run homer off VandenHurk in the sixth increased Colorado's lead to 6-1.
VandenHurk (2-2) was replaced after walking Garrett Atkins. He allowed six runs, five hits, walked three and struck out nine in 5 2/3 innings.
"I just made mistakes, they swung it hard and hit it out," VandenHurk said.
Cantu and Wes Helms hit run-scoring singles in the seventh to cut Colorado's lead to 6-3.
The Rockies added an insurance run in the ninth on Yorvit Torrealba's RBI single.
Closer Huston Street had a four-out save as he retired Cantu on a flyout to left with two runners on base. Street pitched a perfect ninth with two strikeouts for his 29th save in 30 opportunities.
"You make it a lot easier on yourself getting out of that [eighth] inning," Street said. "It gave you a lot a room to breathe."
The Marlins finished with 13 hits in the second game, extending their consecutive games with 10 or more hits to 13. But Florida also stranded 13 runners.
"To win a doubleheader it's hard," Florida manager Fredi Gonzalez said. "We left it all out there in the second game to get back in it."
In the first game, Coghlan's three-run homer followed Volstad's two-run single in the second inning as the Marlins scored five runs off Colorado starter Aaron Cook for a 6-1 lead.
"The team did another great job of scoring runs," Volstad said. "You can't ask for anything more as a pitcher."
The huge advantage was sufficient for Volstad to win his fifth game in six decisions. Volstad (9-9) gave up one run, four hits, walked four and struck out five in five-plus innings.
Cook (10-5) left after surrendering Ross Gload's bases-loaded RBI single in the third. Cook pitched 2 1/3 innings, allowed seven runs, eight hits, walked three and struck out two. He has lost two straight decisions after a streak of seven wins.
The Rockies scored twice in the seventh, cutting Florida's lead to 7-3. Pinch-hitter Ryan Spilborghs hit a run scoring double and later scored on Gonzalez's sacrifice fly off Florida reliever Dan Meyer, who replaced Volstad in the sixth.
Florida scored twice in the bottom of the seventh on Gload's sacrifice fly and Wes Helms' pinch-hit single. Cody Ross' run-scoring single scored Ramirez in the eighth.
"I think hitting is contagious," Coghlan said. "For me, to be able to get some hits in front of Hanley and the guys behind me it's great. Everybody is seeing the ball well."
Marlins 1B Nick Johnson left the first game in the top of the second with a right hamstring strain. Johnson said he was hurt running out a fly in his only at-bat in the first. Gload moved from right field to first base and Jeremy Hermida took over in right. ... Ross went a combined 6 for 9 in the two games, setting a Florida record for hits in a doubleheader. Josh Willingham held the previous record with five July 30, 2006. ... Ramirez's home run off Cook was the first allowed by a Rockies starter in five games.