"I have to be able to establish that right half of the plate to be successful," Washburn said. "When I'm able to do that, I'm able to have pretty good success."
Washburn (1-1) gave up six hits, and improved to 11-3 against the Rays.
"Washburn was extremely solid," Mariners manager John McLaren said. "He did a good job of moving the ball around. It was nice to put some runs on the board to give the guys a little breather."
"We're not the type of team that's going to blow many teams out," Washburn said. "We're going to win a lot of close ballgames with the type of pitching staff that we have."
Tampa Bay right-hander Andy Sonnanstine (1-1) allowed four runs and seven hits over six innings for the Rays, who have lost four straight. Willie Aybar went 3-for-4 with a pair of doubles for Tampa Bay.
"I felt like I gave our team a good chance to win, but it didn't go my way tonight," Sonnanstine said. "Washburn's a good pitcher and he kept our hitters off balance all night. He did a good job."
The Rays have scored seven runs in the past four games.
"We just have to, obviously, swing the bats better," manager Joe Maddon said. "The approach at the plate has got to improve. We have to work better at-bats, and that's going to happen."
Tampa Bay's top five hitters went a combined 1-for-18.
"In this game, success and failure is contagious," Rays designated hitter Jonny Gomes said. "When you see your big dogs failing, it's kind of contagious a little bit."
Burke's first homer of the season, coming off J.P. Howell in the eighth, put the Mariners up 7-1. He finished 2-for-3 on the night, scoring twice.
"He was busting me in to try to get the double play," Burke said. "I was probably guessing more than anything and was able to keep it fair."
Suzuki had an RBI double during a two-run third as Seattle took a 2-0 lead. Tampa Bay got one run back in the bottom of the inning on Akinori Iwamura's sacrifice fly, but Beltre gave the Mariners a 3-1 lead with a solo homer in the fifth inning.
Beltre went 2-for-5 against the Rays, and has a hit in eight of his last nine games.
Seattle extended its advantage to 4-1 on Vidro's run-scoring single in the sixth.
The announced attendance was 12,106, which came one day after the Rays had a turnout of 36,048 for their home opener.
Burke has two career homers. The other came last Sept. 30. ... The entire Tampa Bay roster will wear No. 42 on Jackie Robinson Day on April 15. ... Mariners C Kenji Johjima, mired in an 0-for-19 slide, was rested. ... Rays OF Cliff Floyd will undergo surgery on his right knee Friday and will likely miss four to six weeks. ... Seattle closer J.J. Putz made 61 throws on level ground in his first throwing session since hurting his right ribcage on April 1.