Smoak hit an 0-2 fastball from Bruce Chen (8-4) for his career-best-matching 19th home run, and the Mariners went on to a 4-0 victory, creating a bleak playoff picture for the Royals.
Time is running out -- just five games remaining -- with much distance to be covered and two teams to be hurdled.
The Royals' three competitors for the two American League wild-card spots -- Tampa Bay, Texas and Cleveland -- all won Tuesday.
The Royals, down four games in the loss column to Cleveland, need to win every game and hope that both the Indians and the Rangers completely collapse.
"We just go 'til they say we can't win it," Kansas City manager Ned Yost said. "We go as hard as we can. Keep playing our best guys and staying after it."
Mariners rookie left-hander James Paxton (3-0), making just his fourth major league start, worked a career-high seven innings, allowing four hits, walking none and striking out a career-high 10.
"He looked real good. Big, tall kid, straight over the top, straight downhill action on all his pitches," Yost said. "Good fastball, I mean, overpowering fastball at times, got it up to 97 (mph), lot of life on it. But downhill pitches are tough to hit, especially fastballs. Good breaking stuff, good changeup, good curveball."
Paxton said it didn't matter to him that the Royals had everything on the line.
"I wasn't really paying attention to that. I was just focusing on executing pitches, one pitch at a time," he said. "It could be just the electricity. Being out there, it just kind of raises your intensity up that much more. I think it helps me to get (locked) in."
It was the 13th shutout for the Mariners and the ninth time the Royals have been shut out.
Seattle touched Chen for a run in the first when Brad Miller reached on an infield single followed by Abraham Almonte's single to center. Almonte has reached base in all his 18 games to start his career, 16 with a hit.
With one out, Kendrys Morales singled to left, driving in Miller.
Chen worked five innings, allowing seven hits and four runs. He walked three and struck out five.
The Royals could not respond against Paxton. They put together two singles in the first inning but a double play eased him out of trouble. He didn't allow more than one baserunner in any inning after that, and at one point, he retired 12 straight.
"That guy has a good fastball, slider, change," Royals shortstop Alcides Escobar said. "He got a lot of balls down, run in on the black. Great job for him today. He's sneaky."
The Mariners blew the game open in the fifth with three runs -- all with two outs. Morales, who had three hits, doubled to deep center. Franklin Gutierrez walked, and Smoak hit an 0-2 fastball over the wall in left field.
"I tried to go up with the fastball, and he made good contact," Chen said. "I was ahead in the count. I did not pitch right down the middle. It wasn't a strike. I tried to go up the ladder, and he would chase it and keep my team in the ballgame. (But) he got it.
"I was just trying to do everything I could to help this team win. At that point, it was a 1-0 game. That Paxton did a real good job. It unfortunate because today was a really big game for us."
Smoak has 19 home runs but it was just his second right-handed. The other was Sunday against the Los Angeles Angels.
"I hit it. I don't know if I looked straight into the lights or whatever but I didn't know where it went," Smoak said. "To be able to barrel one up, knowing the work I've put in the last couple weeks, it is starting to show."
The Royals play one more game with Seattle on Wednesday night, then close the season at the Chicago White Sox.
"We're not quitting. We're not giving up," Chen said. "As long as we got a chance, we're going to give it everything we can. Today was big game for us unfortunately. But that's not going to stop us tomorrow."
NOTES: Mariners LHP Danny Hultzen, the second player taken in the 2011 draft, has a problem with his left shoulder and will see Dr. James Andrews on Monday. "Ed (team doctor Ed Khalfayan) saw him and, quite frankly, doesn't like what he sees," GM Jack Zduriencik said. "He saw some damage that he was unhappy with. It's in the shoulder but not in the rotator cuff, more tendon area, labrum." ... The club confirmed Zduriencik will return in 2014. ... The Royals need one more win to match their win total from 1993 (84-78). ... Royals 1B Eric Hosmer had two hits, his ninth multihit game in September. He leads the AL with 59 overall.