Kansas City's bullpen, however, offered the Royals no relief.
Greinke exited after 119 pitches and leading 2-0, but the Mariners batted around in a three-run eighth to beat the Royals 3-2 on Tuesday night.
"Greinke was tough," Mariners manager Don Wakamatsu said. "Getting him out of there maybe before he was ready in a sense just because of pitch counts, allows us to get to their bullpen."
Relievers Jose Rupe (0-1), Robinson Tejeda and Bruce Chen blew it for Greinke and the Royals. The bullpen is 3-6 and has seven blown saves in 13 opportunities, and the highest ERA in the American League at 6.61
"We're throwing away games that we should be winning and probably will end up biting us by the end of the year if we don't stop making mistakes in key situations," Greinke said. "I feel like we've played better than every team so far. We hit better and pitch better, but we'll have one bad inning or one situation where we'll ruin a good opportunity and playing good, but not getting the job done. It's cost us."
Greinke has left three games with a lead only to watch the bullpen give it away. He was 5-0 with a 0.50 ERA in five April starts last season, but was 0-2 with a 2.56 ERA in five starts this year.
"Sometimes when you get Greinke out of there, it's a relief," Wakamatsu said.
Ichiro Suzuki's one-out bunt single started the Mariners' rally in the eighth. Suzuki went 2 for 5 for his seventh consecutive multihit game at Kauffman Stadium, which matches a career high for an opposing batter.
The inning included three walks, Franklin Gutierrez's RBI single, Jose Lopez's two-run double, which was a high chopper off Tejeda's glove that deflected into right-center, and Chen walking Milton Bradley with the bases loaded.
Tejeda, who allowed both inherited runners to score and did not retire a batter, was charged with his second blown save in as many opportunities.
"I'd rather see them banging the fences down in our bullpen rather than walking guys," Royals manager Trey Hillman said. "Three different guys ended up giving up walks in situations where it can't happen. Even with that having happened, we shouldn't have been in that situation."
The Royals went 1 for 14 with runners in scoring position, stranded five runners at third base and had the bases loaded in the first with none out and failed to score.
"Zack deserved much better," Hillman said. "He deserved better offensive support. He deserved a better job out of the bullpen."
Greinke, who threw a one-hitter last August at Seattle, has not allowed a run in his past 29 innings against the Mariners. He is 3-0 in nine career appearances with a 1.86 ERA against Seattle.
The Royals broke through with single runs in the fourth and fifth innings off Seattle starter Ian Snell.
"I got myself out of a couple of jams," Snell said. "I beared down with everything I have in my soul and just went out there and got out of them. I've been doing a lot of soul searching. I went though a lot in a week and had some down time to myself. I kept away from the team a little bit to rethink some things and look at myself and kind of re-strategize how my season is going to go."
Bradley, who had two hits off Greinke and the game-winning RBI with a walk, was booed by the crowd.
"You're supposed to get booed on the road," Bradley said. "It's a badge of honor. If I get a whole crowd of people swaying, moving with my every movement, I've got control out there. I feel like I'm invincible. There's nothing I can't do out there. They give me the energy. If they were being quiet, then I would think something is wrong."
Bradley said Chen tried to set him up by throwing breaking pitches inside before walking on an away pitch with the count full.
"I almost went after it," Bradley said. "But that eagle eye I've got up there took it for a walk for a game-winning RBI."