Gonzalo Pineda also scored on a first-half penalty kick for the Sounders (9-3-2). Pappa's goal was his first of the season, and Barrett scored for the second straight game, giving him three goals for the year. Martins tallied his sixth.
While Seattle won its league-high fifth straight home match, Salt Lake (6-1-6) fell for the first time, missing a chance to set an MLS record for longest unbeaten streak to start the year.
Los Angeles (12-0 in 1996 and 10-0-2 in 2010) and Kansas City (10-0-2 in 2000) previously had 12-game season-opening undefeated streaks.
Pineda's penalty kick was set up when a loose ball in the RSL box was sent toward the left post. Federal Way's Lamar Neagle of the Sounders chased after it, and was pushed to the ground by Aaron Maund, Pineda then beat goalkeeper Jeff Attinella to the back left side midway through the 42nd minute.
Pappa's goal late in the 55th minute came off of a direct free kick, launched over a six-man wall from 23 yards away on the right side.
Barrett made it 3-0 early in the 62nd minute with a 12-yard shot that went high into the net.
Martins notched the final goal late in the 90th minute on a give-and-go with Kenny Cooper.
The Sounders only have one more game before the two week World Cup break. Seattle squares off against Chicago in Illinois on Saturday.