Scott Wedgewood made 24 saves, and Shayne Gostisbehere scored for Arizona (5-18-2), which has lost three straight and five of its past six.
"I thought we played a good first two periods, it got a little squirrelly early in the third and we weathered it, but things happen in this league, and they're good team over there," Gostisbehere said. "`Wedgey' was giving us a chance to win, and we let him down."
Peterson appeared to score at 6:05 of the third period, but the goal was waved off after a video review determined he kicked the puck into the net.
However, Peterson put the Stars ahead 2-1 at 9:01 after Alexander Radulov carried the puck around the net and fed him in front.
"He's got a knack for that, going to the net," Dallas coach Rick Bowness said of Peterson. "He's hanging around the net, he takes the puck behind the net, and he's not afraid to go into the heavy area."
Benn scored on a power play at 15:14 to make it 3-1, then scored again at 17:08 for the 4-1 final.
"He's a big body in front who can make plays and that's what makes him so good," Gostisbehere said. "That's just what he does."
Benn hadn't scored in his previous five games before moving onto a line between Peterson and Radulov.
"It was a great third period," Bowness said. "We had a number of guys in there coughing, hacking, but they sucked it up big time in the third period and we went to work. We'll take that one for sure."