With Tampa Bay leading 7-0 in the seventh inning, Archer hit Kansas City catcher Salvador Perez in the left elbow with a pitch, and Perez took issue with it.
Words were exchanged, providing the emotional highlight of a game that had very few of them, at least from the Royals' point of view.
Archer (3-1) won for the first time in five starts since April 14 to help the Rays break a three-game losing streak. He scattered five singles and walked none, and his control is one of the reasons Perez had a problem with the pitch that hit him.
"Yeah, of course he threw at me. He's going to throw at me because I had two hits against him," Perez said. "I think he was mad. I don't think that's the right way."
"Honestly there was nothing malicious there," Archer said. "I've had some great interactions with him the past. He's a good hitter; I'm trying to pitch inside. There was no malicious intent with 96 mph."
Asked if he understood Perez's angry reaction, Archer said: "No."
Morrison's ninth homer of the season, and his fourth in seven games, came off Kansas City starter Jason Hammel with a man on in the third inning. Weeks followed with his second home run, putting the Rays up 4-0.
Rasmus hit a grand slam off reliever Peter Moylan in the eighth, the sixth slam of his career.
Everyone in the Tampa Bay lineup had at least one hit for the second straight day. Corey Dickerson had a triple and a single and scored three runs.
Hammel (1-4) gave up six earned runs and 13 hits in seven innings, dropping to 0-6 in his last seven road starts.
The Rays scored their first run on a sacrifice fly double play. With Dickerson on third base, Evan Longoria on first and one out in the first inning, Morrison hit a short fly ball that was caught by Royals shortstop Alcides Escobar. Longoria was caught too far off base and doubled up, but not until after Dickerson had tagged up and scored.
Archer threw 101 pitches, marking his 17th straight start in which he threw at least 100 pitches. It was the 20th double-digit strikeout game of his career and his first win against the Royals.
"(The Royals) have given all of us fits and have given (Archer) fits, too," Rays manager Kevin Cash said. "I don't think he did anything differently tonight. He's in a good rhythm right now and it carries over game to game."
FIRST HIT, RBI FOR SOLER
Jorge Soler drove in Kansas City's only run with a ninth-inning double off reliever Jose Alvarado. It was the first hit as a Royal for Soler, who came in a Dec. 7 trade with the Chicago Cubs for reliever Wade Davis. Soler came off the DL on Saturday after having a strained left oblique. He is 1 for 16 in five games.
Royals: RHP Ian Kennedy (hamstring) has resumed throwing off a mound. ... RHP Seth Maness, who underwent surgery on Aug. 18 to repair a torn UCL in his throwing elbow, had his contract purchased from Triple-A Omaha.
Rays: RF Steven Souza Jr. (sore thumb) was out of the lineup for the second consecutive game. ... SS Matt Duffy (Achilles' tendon surgery) had no soreness one day after his first rehab game with Class-A Charlotte. He had Wednesday off.
Thursday's series finale features two of the hottest pitchers in the AL. Royals LHP Jason Vargas is 4-1 with a league-leading 1.19 ERA. Rays RHP Jake Odorizzi (2-1, 2.88) has given up only eight hits in his last four starts, one of which he left after one inning with a strained hamstring.