Cooper scored off a cross from Kalif Alhassan in the 22nd minute.
The expansion Timbers (3-3-1) recorded their first shutout, improving to 3-0-0 at home. Portland was coming off a 3-0 road loss to the Los Angeles Galaxy last weekend.
''We performed unbelievably well tonight and that says a lot after last week's defeat,'' Timbers coach John Spencer said.
RSL (4-1-0) was coming off a demoralizing loss in the CONCACAF Champions League final Wednesday night. The 1-0 loss in the second leg to CF Monterrey of Mexico ended RSL's bid to become the first MLS team to win the tournament and earn a berth in the FIFA Club World Cup. It also snapped a 37-game home winning streak in all competitions.
Real Salt Lake entered the game as the only MLS team without a loss or tie. The club was playing its first league match in nearly three weeks. Goalkeeper Nick Rimando had allowed only one goal all season, but Portland made him work Saturday night.
Rimando lunged and deflected a shot by Jack Jewsbury in the 41st minute. The Timbers had another near miss a few minutes later when Rimando just got a hand in the way of a blast from Alhassan at the upper right corner.
The Timbers had allowed 13 goals coming into the game, most in the MLS. However, goalkeeper Troy Perkins was sharp in his second appearance after missing the start of the season because of a leg injury with three saves.
''We wanted that clean sheet,'' Perkins said. ''We came out organized and we believed we were going to fight until the end.''
Before the match, the Timbers honored longtime fan Gisele Currier, a staple of the Portland professional soccer scene who died earlier in the week at 55. Roses were placed on Currier's usual seat while a bagpiper played ''Amazing Grace.''