Archer (6-6) made a point of saying he didn't appreciate Ortiz staring him down and flipping his bat before beginning his home run trot.
"He's not the right guy to be saying that, I don't think," Ortiz said.
"He's got two days in the league," the slugger said, adding a couple of expletives.
Ortiz's 25th home run came after a walk and Dustin Pedroia's infield single in the third inning, giving Boston a 3-0 lead. It was his fifth homer in seven games and his 29th at Tropicana Field, most among visiting players.
"I don't know what makes him think that he can showboat the way he does, and then nobody retaliates," Archer said. "Nobody looks at him in a funny way or pitches him inside. I don't know why he feels like that, but obviously he feels the way David (Price) said he does -- he feels like he's bigger than the game. He feels like the show is all about him."
Price and Ortiz began feuding after Big Papi homered off the former AL Cy Young winner last fall in the AL Championship Series. Price hit Ortiz with a fastball earlier this season, leading to more verbal sparring.
"He played it up pretty good," Rays manager Joe Maddon said of Ortiz's homer. "The man has meant so much for the game and also back in his country (the Dominican Republic), but there's those moments that occur like that. I'm not saying that my respect diminished, it's just something that makes you think a little bit more."
After hitting his 456th career homer, the 38-year-old Ortiz had the last word. For now, anyway.
"Maybe they get too sensitive about things, just leave it like that," he said. "I think (Archer) is a good pitcher. I think he's got great stuff. He's a guy that I think he's going to be pretty good, but it's going to take some time to get to that level."
The teams next meet in late August at Tampa Bay.
Webster (1-0) pitched 5 1/3 innings in his first game this year in the majors, giving up two runs on three hits and five walks while striking out four.
Koji Uehara pitched the ninth for his 21st save, striking out the last three batters.
The win was the first of the season for the Red Sox at Tropicana Field, where they had lost five straight. Pedroia and Daniel Nava had two hits each for the Red Sox, who did not have a hit over the last five innings.
The Rays' nine-game winning streak was their longest in 10 years and the second-longest in franchise history.
Boston manager John Farrell has talked with backup INF-OF Mike Carp, who recently requested a trade. "I respect his desire to play more, and yet we've been very consistent with his role," Farrell said. "I understand when players want to get on the field more consistently." ... Derrick Brooks, who will be inducted into the NFL Hall of Fame next month, threw the ceremonial first pitch. ... The Rays open a three-game interleague series Monday night with RHP Jake Odorizzi (6-8) pitching against Milwaukee RHP Kyle Lohse (11-4). ... Boston RHP Clay Buchholz (5-6) and Toronto knuckleballer R.A. Dickey (8-10) are the scheduled starters Monday night in the opener of a three-game series.