The Angels amassed 12 hits over the first three innings in taking a 5-0 lead. A two-run double by Rivera made it 7-2 in the ninth, but Baltimore did not go quietly.
Tejada, who earlier hit his 23rd homer, drove a 1-2 pitch over the right-field scoreboard for his ninth career grand slam, the second this season.
The opposite-field shot got Baltimore within a run, but Rodriguez struck out Javy Lopez to earn his 29th save and extend the Angels' lead in the AL West to 3½ games over Oakland, which lost to Detroit 4-1.
"With the bats they have, you know it's never going to be easy," Scioscia said. "They have a lot of power, a lot of guys who know what they're doing. ... They got back into it, brought the tying run to the plate."
Angels starter Jarrod Washburn (7-7) allowed one run, eight hits and one walk in 6 1/3 innings to earn his first win in four starts since July 14. He was 0-3 in that span.
Cabrera, who hadn't had a three-hit game since July 31, had three hits by the third inning. Rivera matched his career high with four hits and threw out two runners trying to stretch singles to left field.
"I always try to keep my arm in shape. I have a good arm, but I don't get many opportunities to put it together," Rivera said through an interpreter. "Sometimes, I try to trick and see if they'll run on me. Tonight, they did."
Robb Quinlan, activated from the 15-day disabled list before the game by the Angels, went 2-for-3 and scored twice. He came in batting .159.
Tejada went 3-for-5 with five RBI and Lopez hit his 11th homer for the Orioles, who have lost four straight.
"The good thing about it was that we had some hits tonight," Orioles interim manager Sam Perlozzo said. "We battled all the way to the end. Give them credit, they hung in there."
Tejada has carried the Orioles for much of the season, but Perlozzo knows Baltimore can't sustain success as a one-man show.
"You never want to put everything on Miggy's back," Perlozzo said. "Miggy can't carry this team, not without somebody else chipping in. Miggy might get you over the hump, but if he's the only guy, that's not going to happen."
The Angels got three hits and a walk in the first inning against Eric DuBose (1-1), but failed to score because Chone Figgins got caught stealing and Garret Anderson hit into an inning-ending double play with the bases loaded.
In the second, Los Angeles used three singles to load the bases with no outs. DuBose retired the next two batters and appeared poised to again escape unscathed, but Cabrera hit a two-run single before Darin Erstad singled in a run.
DuBose, who gave up one hit in six shutout innings in his only other start on Aug. 17 at Oakland, yielded three runs and eight hits in two innings.
"One of the things he was using that was successful last time was his big breaking ball. He used it one time, it got hit and he didn't go to it again," Perlozzo said.
Said DuBose: "I was scuffling right from the get-go. I didn't make the pitches when I had to, and whenever I made a mistake they made me pay."
After DuBose left, the Angels experienced similar success against rookie John Maine in the third. Rivera and Quinlan both singled with one out, and Figgins and Cabrera hit two-out RBI singles for a 5-0 lead.
After his rough start, Maine blanked Los Angeles on one hit over the next four innings.
- Figgins got two hits, his team-high 42nd multihit game.
- Chris Gomez went 2-for-2 against Washburn and is 9-for-21 (.429) lifetime against him.
- Los Angeles is 31-10 in the first game of a series.
- Anderson went 0-for-4 with a walk in his return from an eight-game absence with back and knee injuries.