Neither team has lost in any of its last eight games, those streaks being extended in Wednesday's deadlock.
Darlington Nagbe put Portland in front in the 70th minute by volleying in a cross from Rodney Wallace.
Soon after, however, referee Juan Guzman called a foul on Portland defender Andrew Jean-Baptiste in the box. Cooper converted the penalty, bringing Dallas (6-1-3) a point that kept it six in front at the top of the Western Conference.
Portland moved into second place by itself in the West with a 3-1-6 record. The six ties for Portland are the most this year in MLS.