NHL scoring leader Jamie Benn had three assists for the Stars, giving the 2014-15 Art Ross Trophy winner 15 points this season.
Mike Santorelli, Shawn Horcoff and Carl Hagelin all scored in the final 5:16 of the first period. Frederick Andersen finished with 30 saves, including 15 in the third period when the Ducks were outshot 16-2.
Anaheim, which has lost eight of its first nine games (1-6-2), came in with just six goals in the opening eight games.
Roussel's first goal the season came on the rebound of Vernon Fiddler's shot off the pads Andersen
Defense dominated the first period until the Ducks had their first power-play opportunity. Midway through it, with 5:16 left, Cam Fowler passed to Santorelli, who sent a snap shot from the top of the left faceoff circle past Niemi.
Anaheim had only one goal in the first 22 power plays. Dallas had killed all 15 penalties in the six previous games.
The teams were at even strength when Niemi and defenseman Johnny Oduya both went after Andrew Cogliano to the right of the net. He passed back in front to Horcoff, who had an open net for his first goal since last season when he played for Dallas.
Just 14 seconds later, Niemi played a loose puck behind the net, Alex Goligoski gave it up and Corey Perry passed out to Hagelin for the third goal with 50 seconds to go.
Anaheim outshot Dallas 9-6 in the first.
Sharp, acquired from Chicago during the offseason, scored his first for Dallas on the power play at 7:30 of the second. Benn passed out to Sharp in the slot, and he sent a snap shot past Andersen into the lower left corner.
Eakin scored a short-handed goal, taking a backhand pass from Benn at the edge of the left circle and firing a wrist shot into the net with about 4 1/2 minutes to go. Klingberg's tying goal came about 90 seconds later, when he took a pass from Tyler Seguin in front of the net and beat Andersen.
Ducks captain Ryan Getzlaf has appendicitis and is scheduled to undergo an appendectomy Wednesday in Orange County after feeling intermittent discomfort over the last six weeks, the club said in a statement released Tuesday.
Getzlaf is expected to miss approximately four to 12 days of game action, according to the statement. It is not considered an emergency procedure.
Getzlaf, 30, was in Dallas with the Ducks for their game Tuesday night against the Stars, but he flew back to Southern California on Tuesday to prepare for the procedure.
Like the rest of the Ducks, Getzlaf has struggled on the ice this season. The two-time Canadian Olympic gold medalist had one assist and a minus-5 defensive rating in eight games, and had been credited with 22 shots on goal.
The Ducks finish their current trip Thursday at St. Louis.
Staff writer Helene Elliott contributed to this report from Los Angeles.