Instead, Rivera closed out a stellar performance by Andy Pettitte in a wild ninth inning, making Jason Giambi's homer and Alex Rodriguez's sacrifice fly stand up in the New York Yankees' 2-1 win over the Athletics on Sunday.
"This is what we expect," manager Joe Girardi said of his team's three-game sweep.
The Yankees (53-45) won their seventh straight at home and climbed a season-best eight games over .500 entering a three-game series Monday night against Minnesota. The winning streak at Yankee Stadium is their longest since last September.
The A's, meanwhile, have scored just nine runs while losing five in a row.
Pettitte outpitched All-Star Justin Duchscherer and went eight innings before turning it over to Rivera, who got the first out in the ninth without any trouble.
Then Sweeney singled and Bobby Crosby followed with a high fly to right that gave Bobby Abreu problems as he looked into the sun. He made the grab— momentarily—and then dropped the ball, and first base umpire Hunter Wendelstedt ruled that Abreu never caught it.
The right fielder immediately threw to second trying to catch Sweeney, who had retreated to first base, but the throw sailed over Jeter's head. Alertly backing up the play, Rodriguez cut it off and made the toss to Jeter just in time to force out Sweeney.
"I didn't know what happened," Jeter said. "I was looking at the runner at first when I heard the crowd, and then Bobby airmailed it. Fortunately it all worked out."
Rajai Davis, pinch running for Crosby, tried to steal second moments later, and Jose Molina delivered a strong throw to end it. The save was Rivera's 24th in 24 chances.
"You have to be able to win those kinds of games," said Jeter, who scored the winning run in the 12th inning Saturday when Molina was hit by a pitch with the bases loaded. "You're not going to score 10 runs all the time."
The teams breezed along in 2 hours, 26 minutes on a sweltering 92-degree afternoon, a day after playing 4 hours, 45 minutes. There was a brief drizzle in the third inning, but all it did was ramp up the humidity.
Perhaps accustomed to steamy afternoons back home in Texas, Pettitte (11-7) was sharp from the start. He gave up only four hits, striking out nine without a walk.
"He was threading that outside corner," Oakland manager Bob Geren said. "He pitched like a veteran on the top of his game."
The Yankees struck first when Jeter hit a one-out double that bounced over the center-field wall in the third inning. Abreu followed with a single, and Rodriguez's sacrifice fly to right made it 1-0.
Oakland answered when Jack Hannahan hit a broken-bat double that just stayed fair down the right-field line. Mark Ellis laid down a sacrifice bunt and Sweeney followed with a single.
Giambi's go-ahead homer in the sixth, his 20th of the year, just cleared the 314-foot sign down the right field line. He finished 2-for-2 with a pair of walks.
"Giambi seems like my nemesis," Duchscherer said. "No matter what I throw him, he finds a way to beat me."
Duchscherer (10-6), who's been battling a virus for about a week and was so ill during Tuesday's All-Star game that he could barely pitch, gave Oakland a chance. He lasted seven innings and gave up only seven hits.
The American League's ERA leader still hasn't allowed more than three earned runs in a game this season.
"It's part of the game. Sometimes you just have to credit the other team," Duchscherer said. "Pettitte threw really well. If he throws like that against any other team it's going to be hard to score runs."