"I'm totally confident in the shootout," said Jokinen, who converted his sixth shootout attempt in as many tries in the second round to end the game.
Turco stopped Dennis Wideman with a sprawling second-round save. Jokinen then beat Sanford with a wrist shot to end it.
The Blues came back to tie it at 2 on a Sillinger goal early in the third period and then grabbed a 3-2 lead with less than seven minutes left on Kevin Dallman's goal.
"We battled back hard at the end, but shootouts are just a roll of the dice," Sanford said.
The Stars have used the six extra points from the shootouts to cement their lead in the Pacific Division.
"I think shootout points are important points," Jokinen said. "We have six points and those will be big at the end of the season."
The Stars put in extra practice time on the shootout, and it's paying dividends.
"It's something we take very seriously," Stars coach Dave Tippett said. "We've talked about that since the first day in training camp."
Dallman put the Blues in front 3-2 on a power-play goal with 6:56 left in regulation, beating Turco on a one-timer from the left circle. But Jokinen set up Dallas defenseman Philippe Boucher's goal with 46.4 seconds to play to send the game into overtime.
St. Louis failed to score on a power-play opportunity in the extra five minutes, and the game went to the shootout. The Blues, last in the league with 30 points, dropped to 1-5 in shootouts.
"They're a hard-working team," Turco said. "They don't have much talent but their goalie played great. He made some unreal saves. But we got the win and we're excited about it."
Zubov put the Stars in front 2-1 on a power-play goal with 9.9 seconds left in the second period, beating Sanford on a 40-foot blast. Brenden Morrow, who also had a goal, provided a screen for Zubov.
Dallas, which had converted only two of its previous 30 power-play chances, capitalized with Sillinger in the penalty box for hooking.
Zubov's eighth goal gave him 40 points for the 10th straight season, the longest active streak among NHL defensemen.
But Sillinger atoned for the penalty with his 20th goal 43 seconds into the third period to pull the Blues even at 2.
The Blues are 1-0-3 in their last four games, extending their season-long streak of games with at least one point.
Dallas dominated the first period, building a 21-3 shots advantage, but the Stars needed Morrow's rebound goal at 17:25 to tie it at 1.
St. Louis got the first goal on its first shot when Wideman beat Turco from the slot on a power play at 3:27 of the first period.
Sanford, who won a shutout over Vancouver on Monday night, finished with 38 saves.
"Sanford was phenomenal," Blues forward Dean McAmmond said. "He kept us in the game and he's been doing that of late. We
should have come away with two points."
The Stars had the most shots in any period this season with their first-period effort. ... Dallas defenseman Trevor Daley was scratched with the flu. Defenseman Janne Niinimaa was also out for the Stars with a sore ankle. ... Patrick Traverse, called up on Tuesday from Dallas' minor-league affiliate, the Iowa Stars, was in the defensive rotation. ... St. Louis lost its sixth straight to Dallas.