The eight-game road winning streak is the third longest to begin a season in NHL history; the Buffalo Sabres won their first 10 games away from home in 2006-07, and the New Jersey Devils won their first nine in 2009-10.
Ryan Hartman, Ryan Johansen, Yannick Weber and Roman Josi scored, and Juuse Saros made 20 saves for Nashville (13-3-0), which trailed 3-1 in the second period and has won five straight.
WATCH: [All Predators vs. Stars highlights]
"We just don't give up," Ekholm said. "You can see it today, 3-1, they're close to getting 4-1. We just dig one out pretty much."
Tyler Pitlick, Tyler Seguin, Jason Spezza and
Denis Gurianov
scored for the Stars (9-6-2), who are 1-1-2 in their past four games. Ben Bishop made 29 saves.
Ekholm scored at 2:27 of overtime, 41 seconds after leaving the penalty box. He was penalized for cross checking at 19:46 of the third period.
"I guess a little bit of karma coming into that," Ekholm said. "Good job by the guys killing it. I was just determined when we got out there."