Dallas (38-26-8) has 84 points, tied with the Colorado Avalanche, who hold the first wild card into the Stanley Cup Playoffs from the Western Conference because they have played one fewer game. The Stars also are tied with the Anaheim Ducks but hold the second wild card because they have more regulation and overtime wins (34-32).
"There isn't any," Stars coach Ken Hitchcock said when asked about his level of frustration. "This is a great point. To come back, and we've got nothing left in the tank and to keep pushing, this is a good point."
Craig Anderson made 23 saves for the Senators (26-33-11), who won their third straight.
"We're showing what kind of team we really are in here," Duchene said. "It's too bad it's too late in the season and we couldn't have done this a little earlier in the year, but it wasn't for a lack of trying. We want to go into the summer on a high note and we're on our way."
Tyler Seguin one-timed a pass from Jamie Benn in the slot on the power play at 10:36 of the first period to give the Stars a 1-0 lead. It was Seguin's 38th goal of the season, a new NHL career high.
Ottawa is 17 points behind the New Jersey Devils for the second wild card from the Eastern Conference.