Kevin Shattenkirk and Barret Jackman also scored for St. Louis, which made another major move to strengthen its defense during the game.
During the first intermission, the Blues announced the acquisition of defenseman Jay Bouwmeester from Calgary in exchange for a first-round pick and two prospects - defenseman Mark Cundari and goaltender Reto Berra.
The move comes 48 hours after St. Louis traded for defenseman Jordan Leopold, who played his first game in a Blues sweater Monday.
Dany Heatley scored for the Wild, whose home winning streak ended at four games. Minnesota lost for just the sixth time in its last 21 games overall.
St. Louis began the day out of the playoffs and staring at a stretch in which it plays six of the next seven on the road. Bouwmeester is expected to be on the ice Thursday against Chicago, the top team in the Western Conference.
Blues goalie Jaroslav Halak left the game with 12.5 seconds to go in the first period after sprawling to try and make a save on a slap shot. Halak had missed time previously this season with a groin injury.
Elliott, fresh off a two-game conditioning assignment in the minors, stepped in and stopped 19 of the 20 shots he faced, including all 14 in the third period.
Elliott was especially strong early in the third when the Wild unleashed six shots in the opening five minutes.
On the other end, Wild goalie Niklas Backstrom's career-best and team-record eight-game winning streak ended.
Schwartz got a clean look from the right circle midway through the first period and beat Backstrom over his right glove.
In the second, McDonald put his own rebound past Backstrom to make it 2-1 at 12:17. Less than two minutes later, Shattenkirk sent a shot from the point through traffic that made it 3-1.
Led by free-agent signees Zach Parise and Ryan Suter, as well as captain Mikko Koivu, the Wild scored four or more goals in six of their last nine games entering Monday and put themselves in the hunt for the Northwest Division title.
But the exciting trio was kept off the board Monday and the rest of the Wild were unable to pick up the slack.
NOTES: If St. Louis does not qualify for this year's playoffs, the Flames will receive St. Louis' fourth-round selection in this summer's draft, with the first-round pick being deferred to 2014. ... Koivu missed time in the first period after getting hit in the face with Chris Stewart's stick. ... Wild forward Matt Cullen was injured late in the second period and did not play in the third. ... Jake Allen was assigned to AHL Peoria to make room for Elliott. ... The Blues also announced that the Vancouver Canucks have agreed to buy Peoria Rivermen, pending approval from the AHL.
Kennedy (1-0) allowed two runs on five hits with one walk.
St. Louis' Adam Wainwright (0-1) went six innings, giving up four runs, three earned, on 11 hits. He struck out six with no walks.
Arizona's Gerardo Parra matched his career best with four hits, three of them doubles. Rookie A.J. Pollock was 3 for 4, including a two-run double, and Martin Prado doubled twice with an RBI and two runs scored for the Diamondbacks.
Matt Holliday had an RBI double and Daniel Descalso a run-scoring single for St. Louis.
The Diamondbacks' seven doubles were more than they had in any game in the last two seasons.
Kennedy, 15-12 last year after going 21-4 in 2011, dominated after giving up consecutive one-out doubles to Matt Carpenter and Holliday to put St. Louis up 1-0 in the first.
David Hernandez threw a perfect eighth and Brad Ziegler did the same in the ninth for the Diamondbacks.
Arizona scored three times off Wainwright in the fourth.
After one-out singles by Miguel Montero and Paul Goldschmidt, Jason Kubel lofted an RBI double to right-center to tie it at 1. Pollock, starting because of injuries to Adam Eaton and Cody Ross, lined a double off the glove of center fielder Jon Jay to bring in two and Arizona led 3-1.
The Diamondbacks added an unearned run on second baseman Descalso's throwing error in the fifth. Prado, acquired from Atlanta in the Justin Upton trade, doubled to deep right-center, then Aaron Hill hit a slow grounder to second. Descalso threw in the dirt trying to throw to first and the ball bounced off the glove of first baseman Allen Craig as Prado raced home to make it 4-1.
Arizona got two more in the seventh. With Fernando Salas on the mound, Parra and Prado doubled, then Hill singled. Marc Rzepczynski relieved Salas and gave up an RBI sacrifice fly to Montero.
Notes: Entering the game, Kennedy was 1-3 against Cardinals with an 8.59 ERA. ... Kennedy was the winning pitcher when Arizona beat San Francisco in its season opener a year ago. .. Former Diamondbacks ace Brandon Webb, whose career was cut short by arm trouble, threw out the ceremonial first pitch. ... Wainwright made his second career opening day start. ... The Cardinals send left-hander Jaime Garcia to the mound Tuesday night while the Diamondbacks go with right-hander Trevor Cahill. ... The only other Arizona rookie with three hits on opening day was Travis Lee, who did it in the Diamondbacks' first game in 1998. ... Attendance was a sellout of 48,033.
Manchester police and the FBI are asking for help identifying a man they believe spiked the soup the Dierbergs at Lafayette Center. Employees pulled the soup container on Thursday, after witnesses saw a man pouring a foreign substance into the container near the salad bar. The soup was removed before any was purchased. The soup-spiker was caught on surveillance video before he could make his escape. He's described as a Caucasian man between 50 and 60 years old with a heavy build and light beard.
A standing-room only crowd will have to come back another night to see the conclusion of Ellisville Mayor Adam Paul's impeachment hearing. Proceedings lasted for a little more than six hours last night. Hundreds of residents and spectators packed the Ellisville Elementary School gymnasium, which was chosen to host the event because of the expected size of the crowd. Ellisville City Counsel members are seeking to remove the 32-year-old Mayor for office, claiming he abused and overreached his authority. Paul's lawyer Chet Pleban unsuccessfully tried several methods to get the hearing thrown out. Pleban says the counsel is biased and the lawyer for the prosecution was illegally chosen. All of Pleban's objections were overruled. The hearing is scheduled to resume tonight at 7:00.