Perron skated in on Jets goalie Chris Mason and sent a backhander high over his outstretched right pad and into the upper corner of the net, handing the Blues their fourth win in six games.
Perron also scored a regulation goal and David Backes found the net as well for St. Louis, which holds down second place in the Central Division.
Blake Wheeler scored a pair of second-period goals for the Jets, who came into the game with a one-point lead in the Southeast Division.
Mason finished with 14 saves for Winnipeg.
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