Elliott stopped 19 shots, and defenseman Kris Russell scored in his St. Louis debut to lead the Blues to a 3-0 win over the Lightning on Saturday.
St. Louis improved to 5-0-1 in its last six home games. The Blues have earned five points in three games under new coach Ken Hitchcock, who replaced Davis Payne on Sunday.
Tampa Bay had a three-game winning streak snapped.
Elliott (6-1) recorded his second shutout of the season and 11th of his NHL career. He has two shutouts in his last four starts. Elliott made 32 saves in a 3-0 win over Vancouver on Oct. 26. He stopped 11 shots in the third period on Saturday and kept the Lightning in check during a frantic final minute.
Elliott and his teammates had been searching for a solid performance throughout the lineup, and they got it.
"That was the complete game we've been wanting," Elliott said. "Everybody was hustling. Everybody was working hard. No one took a shift off."
Shattenkirk converted off a 2-on-1 break late in the second to make it 2-0. The defenseman has two goals and is tied for the team lead with 11 points.
Oshie, who scored the Blues' first short-handed goal, has five goals this season and four in five games.
Elliott faced just eight shots over the first two periods but took over down the stretch.
"He was tremendous," Shattenkirk said. "He's battled for us all year. We're getting used to him in there saving us."
St. Louis center Alex Steen said the duo of Elliott and Jaroslav Halak have been the key to the Blues' 8-7-1 start.
"As a one-two punch, they've been great," Steen said. "[Elliott] made some key saves in key moments."
Russell performed well in his St. Louis debut. He got the ball rolling with an early goal after a cross-ice pass from Evgeny Grachev.
"I was fortunate," Russell said. "We had bodies in front, and I just wanted to get it on net."
Tampa Bay goalie Mathieu Garon fell to 3-3-1. The Lightning, who had a season low in shots, were shut out for the first time this season.
Lightning coach Guy Boucher wasn't pleased with his team's performance. Tampa Bay had won four of five and seven of nine.
"The first and second periods, sloppy all the way," Boucher said. "That's the story of the game."
He didn't get any argument from his players.
"I don't think we played well enough," defenseman Eric Brewer said. "They kept moving forward much more than we did."
Hitchcock improved to 2-0-1 in his first week with St. Louis.
"There are things that we are still learning to do," he said, "But tonight, we managed the game properly. We didn't get frustrated when we didn't score."
- Tampa Bay G Dwayne Roloson didn't play despite being 7-2-1 with a 1.58 goals-against average in his last 10 starts against St. Louis.
- Tampa Bay has been outscored 19-11 in the first period this season.
- The Blues have scored at least one third-period goal in the last nine games.
- Shattenkirk had a team-high four shots on goal.