The Ducks trailed 3-1 heading into the third period but pulled out a victory that brought them a vital two points in their bid to earn a spot in the playoffs.
Anaheim, Dallas and Nashville are tied for the eighth and final post-season berth with 75 points. They are only three points behind fourth-place Chicago in the compacted Western Conference race.
Visnovsky scored Anaheim's first goal midway through the first period but Dallas scored three times in the second, appearing to take control.
Visnovsky began the Anaheim comeback 5:24 into the third with a power-play goal and Teemu Selanne tied it with 1:03 left in regulation.
Dallas center Tom Wandell was sent to the penalty box for hooking with 25 seconds to go in the extra period and Visnovsky capitalized by scoring on a slap shot off a faceoff.
Dan Ellis made 23 saves to get the win.
All four Anaheim goals were allowed by Kari Lehtonen on 29 shots.