St. Louis had lost its first two shootouts this season, but after rallying twice during regulation, the Blues pulled out the extra point with the help of Conklin's game-ending stop on Robidas.
"I kind of guessed he was going to shoot low," Conklin said after the Blues improved to 3-1 in their last four after sinking to the bottom of the Central Division. "This game was pretty important for the position we're in."
Brad Richards scored for Dallas in the first round, but Andy McDonald countered for St. Louis. Dallas' Mike Ribeiro hit the goal post with his second-round attempt, and Boyes scored through Marty Turco's pads to give the Blues the edge.
The Stars are 1-5 in shootouts this season.
"It's a big part of the game and we do put emphasis on it in practice," Dallas' James Neal said after returning from a two-game league suspension. "We've got to bear down and find a way to win that extra point. It's so big."
Erik Johnson scored a first-period power-play goal to snap St. Louis' 0-for-19 drought with the man-advantage. The Blues started the night last in the league on the power play at over 12 percent.
"Our power play hasn't been clicking lately but the attitude tonight was to work hard and get the pucks and play with a 5-on-5 mentality," said T.J. Oshie, who had put the Blues in front midway through the final period.
David Backes added a goal and an assist in regulation, and Conklin stopped 34 shots.
Oshie gave the Blues a 3-2 advantage with 8:59 remaining, firing a wrist shot from the right circle that beat Turco on the glove side for St. Louis' first lead.
But Brenden Morrow tied it at 3 for Dallas on a deflection off Richards' shot with 25.4 seconds left in regulation and Turco off for a sixth attacker.
Robidas scored on the power play, and Toby Petersen added a goal for the Stars, who got a 35-save performance from Turco.
Dallas had the game's first nine shots and took the lead on Robidas' one-timer at from the left point at 5:37.
But the Stars couldn't maintain that early momentum and St. Louis answered at 9:00 of the first period as Johnson fired into an open net with the man advantage to beat an out-of-position Turco.
"It was a sloppy game. We started out doing what we wanted to do, but for some reason we decided not to do it anymore," Richards said. "That's not how we wanted to play. We started turning pucks over and not doing what we did in the first 10 minutes. We weren't going to keep them to zero shots for the whole game."
Petersen's one-timer from the right circle got through Conklin's pads 1:43 after Johnson's goal to give Dallas a 2-1 edge.
Backes' rebound at 10:02 of the first drew St. Louis even at 2.
Neal, Dallas' top goal-scorer with 11, was back on the top line with Richards and Loui Eriksson. The Stars had won both games with Neal out, and his teammates kidded him plenty about that during his suspension.
"It felt good coming back," Neal said. "I stepped right back in. But it's never fun when you step back in and the guys lose."
The Stars honored team executive and former player Brett Hull in pregame ceremonies recognizing Hull's recent induction into the Hockey Hall of Fame. Team owner Tom Hicks announced that Hull, who walked onto the ice holding the Stanley Cup, will become the first member of the Stars Walk of Legends. Hull played for the Blues from 1987-98 before signing as a free agent with the Stars, helping Dallas win the 1999 Cup. Hull retired with 741 career goals for Calgary, St. Louis, Dallas and Detroit before retiring in 2005. ... The Blues are without LW D.J. King (hand). ...Richards extended his points streak to 10 games. ... Dallas D Nicklas Grossman missed his second straight game due to a lower-body injury.