Both Wild Card leaders, Tampa Bay and Cleveland, won their fifth straight games. Texas stayed one game behind Cleveland for the second spot.
"We just go until they say we can't win it," Royals manager Ned Yost said. "We go as hard as we can. Keep playing our best guys and stay after it."
Elimination could come as soon as Wednesday night if the Royals lose and the Indians win.
Justin Smoak's three-run homer gave the Mariners a 4-0 lead against left-hander Bruce Chen in the fifth inning as light rain began to fall and the stadium roof was closed.
There were two outs when Kendrys Morales launched a drive to deep center field that flicked off the glove of a galloping Lorenzo Cain for a double.
"I should have caught it. It tipped my glove and if it hits my glove, I've got to catch it," Cain said. "It didn't happen and they ended up scoring three runs."
Franklin Gutierrez walked and Chen got an 0-2 count on Smoak. But the next pitch was re-directed over the left-field wall.
"I was ahead in the count. It wasn't a pitch right down the middle, it wasn't a strike," Chen said. "I tried to go up the ladder and maybe he'd chase it and I'd keep my team in the ballgame."
Problem was, as Yost saw it, Chen didn't go up the ladder quite high enough.
"On 0-2, he tried to get it up and get him to chase a high fastball and he just didn't get it up enough. He just got it up enough to hit it out of the park," Yost said.
It was Smoak's 19th homer this season, but just his second from the right-handed side of the plate.
The Mariners got a first-inning run on three singles by Brad Miller, Abraham Almonte and Morales against Chen.
"You know all of his pitches seem to come out the same, really, they all look the same coming out of his hand," Smoak said. "Fastball curveball, cutter, changeup. … With a guy like him, he knows how to pitch and you've just got to grind out at-bats and try to get something over the middle of the plate, and not miss it. I think we showed that tonight, having good at-bats against him."
Chen was pulled after the fifth inning in favor of rookie left-hander Chris Dwyer, making his Major League debut. Dwyer immediately found himself in a jam -- two singles, a stolen base and an error put runners at second and third with no outs. But he pitched out of it with two strikeouts and a popup.
"I felt good, I felt rested," Dwyer said. "I was anxious when I first got out there, but I knew if I'd settle down and try to get a couple of outs there, everything would work out."
It did and his second inning also was scoreless with a walk eliminated by a double-play grounder.
"He did a nice job," Yost said. "Second and third and pitched himself out of it which was nice. Got through two innings. You look for spots for get young guys in, it was a good spot and he responded nicely."
Meanwhile, the Royals were getting their first look at Paxton and they didn't much like what they saw. He wasn't giving up much of anything.
A Canadian, Paxton seemed to prosper in the 58-degree (and dropping) temperature. He survived singles by Alex Gordon and Eric Hosmer in the first inning. Between Cain's double in the second inning and Emilio Bonifacio's double in the sixth, Paxton retired 12 consecutive batters.
"He looked real good," Yost said. "Big tall kid, straight over the top, straight downhill action on all his pitches. Good fastball -- overpowering fastball at times, got it up to 97, a lot of life on it. Downhill pitches are tough to hit, especially fastballs. Good breaking stuff, good changeup, good curveball."
Paxton worked seven innings and held the Royals scoreless on four hits. He walked none and piled up 10 strikeouts. This was his fourth start since being called up from Triple-A Tacoma and he stands 3-0 with a 1.50 ERA. Nice getaway for the 24-year-old from Lander, B.C., and the University of Kentucky.
"He's got good stuff, he was very firm," Cain said. "Pretty good changeup, stayed inside and had a nice little cutter. Just kind of able to ramp it up and throw 91 to 97, so it's tough to get a key on him. He's a pretty good pitcher, he's got a bright future."
The Royals could generate nothing against relievers Yoervis Medina and Danny Farquhar either and were shut out for the ninth time.
Now, after making a spirited albeit long-shot run for their first postseason berth since 1985, they're on the brink of elimination.
"We're not going to quit, we're not going to give up," Chen said. "As long as we have a chance, we're going to go out there and do everything we can. Today was a very big game for us. Unfortunately, we couldn't do it, but that's not going to stop us tomorrow."