The Blues won for the sixth time in seven games and gained points for a franchise-record 21st straight home game. They picked up their league-best 26th home win (26-3-4).
Minnesota is winless in its past seven and has scored just nine goals during its latest slump.
Jackman had held the longest active scoreless skid in the NHL. Anaheim's Sheldon Bookbank broke a 167-game scoreless streak with a goal Friday.
Jackman sent a shot from the point past goalie Josh Harding for his first goal since Jan. 7, 2010. It gave the Blues a 2-0 lead early in the second period.
"It's not like anyone is expecting me to score," said Jackman, who has 20 goals in his 10-year career. "But it's nice anytime you can bounce one in. I finally got a bounce my way. The guys are all pretty excited for me."
Elliott recorded his sixth shutout of the season and 15th of his career. He improved to 20-5-2 and sports an NHL-best 1.55 goals-against average.
But the day belonged to Jackman, who took plenty of good-natured ribbing from his teammates during the lengthy skid.
"It was awesome to see a guy like that score," Elliott said. "Now, I think there is more to come."
Blues coach Ken Hitchcock was elated to see several of his players break scoring skids.
Reaves, who has nine fighting majors in 40 games, scored for the first time since Jan. 13, 2011. He converted from in front of the net off a pass from Scott Nichol for a 1-0 lead midway through the first period.
"It's about time, it's been a while," Reaves said. "I don't even know how to celebrate anymore."
McDonald tallied for the second time in four games since returning to the lineup last Sunday after missing 51 games due to a concussion. Sobotka scored for the first time in 28 games.
"The bench was so excited because these guys are warriors for us," Hitchcock said. "They've done all the dirty work, all the grinding for us."
Kevin Shattenkirk and Alex Pietrangelo had two assists apiece for St. Louis, which moved within three points of Central-leading Detroit. David Perron added a third-period assist to push his career-best point streak to seven games.
Elliott faced just three shots in each of the last two periods. His teammates blocked 11 shots.
"We definitely didn't give them a chance to get into it," Elliott said. "It's amazing watching these guys. Everybody is committed to coming back and helping out with the (defense)."
The Wild had a season-low in shots on goal, and have struggled offensively in the last month.
"It's pretty tough to pinpoint what it is," Minnesota coach Mike Yeo said. "We came out with the right mindset. We started out OK. The first period wasn't bad. Then we let down in the second."
The Wild got off to a 20-7-3 start this season but have won just five games since. They have scored one goal or less in five of the last seven games.
"This is not where we want to be," Yeo said. "This is certainly something that we have to address."
Blues forward Patrik Berglund failed on a penalty shot early in the second period. Berglund missed high with a backhand.
"I've got to be more responsible with my stick," Peters said. "There was in no way malicious intent to go to his head. But my stick made contact with the side of his head so I'll take the consequences."
Backes was fine after the game and should be able to play Sunday in Chicago.
Linesman Dan Schachte worked his 2,000th game. He will retire at the end of the season. "It's not really a place for a 54-year-old guys anymore," he said. Schachte worked his first game on Oct 4, 1982, at Chicago Stadium. He met his future wife, Kim, in St. Louis. Schachte was given signed jerseys from both teams during a ceremony before the game. ... The Wild have used a league-high 13 rookies this season. ... St. Louis G Jaroslav Halak was scratched from the lineup with flu-like symptoms. Ben Bishop was recalled from Peoria of the AHL to serve as backup. ... The Blues will make their first national television appearance since Jan. 13, 2007, when they face Chicago on Sunday on NBC. ... Pietrangelo, who assisted on McDonald's goal, has points in 14 of his last 20 games.