Captain Jack Jewsbury and Rodney Wallace also scored for the Timbers (2-2-1), playing for the second time in three days before a sellout crowd at their renovated downtown stadium.
Kalif Alhassan had two assists for Portland.
David Ferreira and Brek Shea each had late goals for FC Dallas (1-3-1) to avoid the shutout.
The Timbers were coming off a 4-2 victory — their first in Major League Soccer — over the Chicago Fire on Thursday for Portland's home opener at Jeld-Wen Field. The scoring flurry came despite a steady downpour.
The Timbers started their inaugural MLS season on the road while Jeld-Wen Field was made more soccer friendly.
FC Dallas was coming off a 3-0 victory over Colorado on April 8, also their first win of the season. It was a rematch of the pairing that played in the 2010 MLS Cup, which the Rapids won.
The Timbers went ahead in the 13th minute when captain Jack Jewsbury's kick hit the post then bounced off FC Dallas goalkeeper Kevin Hartman and into the net.
Jewsbury was playing in his 200th MLS match.
Cooper, a Dallas native, headed in a corner kick from Jewsbury in the 35th minute.
Cooper played for FC Dallas from 2006-09 and is still ranked second on the club's all-time goals list with 40.
He signed with the Timbers after spending the last two seasons in Germany, where he was limited by injury.
Cooper also scored the Timbers' first-ever MLS goal in the team's opener at Colorado.
Wallace added the third goal in the 55th minute, taking a running pass from Alhassan and beating Hartman from out front. Alhassan also earned an assist on Jewsbury's goal.
Ferreira, the reigning MLS MVP, scored in the 82nd minute then assisted on Shea's goal some three minutes later.
Ferreira was the MLS player of the week after his two goals in the victory over the Rapids.
New Zealander Jake Gleeson, who started out the season as Portland's third-string goalie, made his third MLS start for the Timbers while veteran Troy Perkins and backup Adin Brown continue to nurse injuries.