Jere Lehtinen and Philippe Boucher also scored and Eric Lindros had two assists for the Stars, winners of three of their last four.
Klemm, moved from defence to right wing by coach Dave Tippett earlier this week, was scoreless in his first 11 games this season.
In the teams' last meeting on Dec. 8, the Oilers came away with a rare win in Texas, 2-0. With the latest loss, the Oilers dropped to 4-20-2 in regular-season games in Dallas since the Stars moved from Minnesota in 1993.
Marc-Andre Bergeron and Joffrey Lupul scored power-play goals for the Oilers, who've lost five of their last seven. Dwayne Roloson had 23 saves for Edmonton.
Aggressive forechecking by Lindros led to Klemm's first-period goal. Lindros forced Edmonton defenceman Ladislav Smid into a turnover behind the Oilers' net and Lindros passed to Klemm, who skated up the right wing and notched his first goal of the season at 10:44 of the opening period.
Lehtinen made it 2-0 at 6:02 of the second period, converting Jussi Jokinen's centring pass.
Boucher extended Dallas' advantage to 3-0 less than two minutes later when he scored his 12th of the season from beyond the right circle.
Bergeron's goal while the Oilers skated five-on-three in the second period cut the deficit to 3-1. Edmonton broke a drought of six games and 30 opportunities with the man advantage. Bergeron registered the Oilers' last power-play tally on Dec. 8 against the Stars.
Lupul made it 3-2 when he lifted a power-play rebound over a fallen Turco with 30 seconds left in the second period.
The Stars remained without centre and offensive catalyst Mike Modano (hip flexor/groin injury), who missed his ninth straight game. Dallas is 5-4 since Modano's injury.
Notes: Matthew Barnaby's assist on Klemm's goal was his 300th point. ... Still out for the Oilers were Ryan Smyth (fractured right thumb, missing his 10th straight game) and Ethan Moreau (dislocated right shoulder, sitting out his 27th in a row).