"This was obviously a game we needed," said Morneau, who was batting cleanup.
Morneau is making a compelling case to be an All-Star selection, which would be all the more special because it's in Minnesota where he began his career. Morneau is hitting .304 with 13 home runs and providing valuable protection for Tulowitzki.
"If I don't get it done, he seems to get it done," Tulowitzki said. "He's been a nice addition to our ball club. Without a doubt, he's All-Star worthy."
Jorge De La Rosa (7-6) allowed five runs, four earned, in seven innings. He is the first Rockies starter to earn a win since June 12.
Nick Greenwood (1-1) pitched 1 2/3 innings in relief of starter Shelby Miller, who left in the third with back tightness but believes he will be back in time for his next start. Greenwood surrendered Morneau's home run on a hanging slider in the fourth.
Morneau had an RBI single in the first and a two-run single in the seventh that gave Colorado some breathing room.
"He's having a heck of a season," Rockies manager Walt Weiss said.
The Rockies finished with 17 hits one night after managing just three when they were shut out at home for the first time in nearly a year.
Jhonny Peralta hit a two-run homer for St. Louis and Matt Holliday had another productive day at Coors Field against his former team, going 2 for 5. He is hitting .545 in his last five games at this park.
De La Rosa finished strong, striking out Allen Craig to end the seventh with Holliday on first.
"I thought he had no shot at that ball," Weiss said. "Great play by Brandon. He's not afraid to get dirty. He's not afraid to leave his feet."
The Cardinals scored two runs courtesy of crazy bounces. In the fourth, a hustling Peter Bourjos scored from second on Tony Cruz's high bouncer off the plate to second baseman DJ LeMahieu. An inning later, a slow roller possibly grazed Holliday's arm -- replays were inconclusive -- as he sprinted toward third, the ball taking a funny hop past Tulowitzki and rolling into left field.
Holliday easily ran home as Tulowitzki protested. Weiss came out to argue the ruling, but the play wasn't reviewed and the run stood to cut Colorado's lead to 6-5.
The Cardinals' rotation appeared to take another hit with Miller leaving the game. The right-hander arched his back in discomfort soon after delivering a pitch to De La Rosa.
Manager Mike Matheny and a trainer instantly trotted out to check on Miller, who took a few warm-up tosses before being taken out for Greenwood.
"I could still throw the baseball. It was just kind of a nagging little sting," explained Miller, who went 2 2-3 innings and surrendered three runs. "But nothing that we can't come back from.
"At the end of the day, I didn't really execute pitches at all. So, obviously, it was disappointing and frustrating."
Cardinals LHP Marco Gonzales will make his major league debut Wednesday. Gonzales was a high school standout in nearby Fort Collins, and his dad, Frank Gonzales, is a pitching coach in the Rockies' minor league system. ... Rockies RHP Christian Bergman has a broken left hand and will not make his start Wednesday. The Rockies are expected to bring up LHP Yohan Flande from Triple-A Colorado Springs.