In the first period, Shawn Horcoff beat Blackhawks goalie Corey Crawford to a loose puck behind the net and passed to Peverley skating down the slot.
Crawford recovered from that goal, stopped Alex Chiasson on a penalty shot early in the second and blanked Dallas the rest of the way en route to his 100th NHL win.
"It's a key situation in the game. It's important if we go down by two goals. I was just trying to come up with a save and keep our team in it," said Crawford, who finished with 28 saves.
Crawford moved toward the left post, leaving Chiasson little room. His shot caromed off the post.
"With the score 1-0, that penalty shot could have put them away. If you lose like that and come away with a point, you can take something from it," Ruff said. "To use a football analogy, we got inside the red zone a few times, but didn't put it away."
Crawford and Lehtonen combined to stop 16 consecutive shootout attempts until Smith found the net.
The game was a defensive struggle.
Neither team put a shot on goal until 2 minutes into the game. Lehtonen later stopped point-blank attempts by Brent Seabrook and Smith.
Dallas' best scoring chance was on a backhand by Jamie Benn at the 13-minute mark until Peverley put the Stars ahead.
Dallas had an 11-9 advantage in shots during the first period.
After the penalty shot, the Stars had another good chance 10½ minutes into the second. Crawford reached up to catch Ryan Garbutt's drive before it reached the upper right corner of the net.
"We needed to get a break, but they got the break on (Sharp's) goal. It tipped off a stick and went in another direction," Ruff said.
No one went to the penalty box until Peverley was called for tripping at 11:31 of the second.
Each team had one shot on goal in overtime, which ended with Dallas shooting a puck wide right.
NOTES: Sharp's goal was his fifth point in two games against the Stars this season. He had four assists at Dallas on Nov. 9. ... At times, a large contingent of Chicago fans in the sold-out arena chanted "Let's go Hawks!"