The vote was a nonbinding straw poll and has no bearing on the selection of the state’s 43 delegates. Of those, 40 are up for grabs, but they will not be picked until later.
All four candidates remaining in the race — Mr. Romney, Rick Santorum, Representative Ron Paul and Newt Gingrich — actively campaigned in Washington, but none concentrated their efforts there as intensely as Mr. Paul.
The victory gives Mr. Romney some momentum heading into the big contests this week on Super Tuesday, when 10 states vote. With 81 percent of the Washington votes counted on Saturday night, Mr. Romney had won about 37 percent, with Mr. Paul at 25 percent, Mr. Santorum at 24 percent and Mr. Gingrich at 11 percent.