Jepsen inherited runners on second base and third base with one out when Weaver departed the game Saturday, but the 25-year-old right-hander retired Michael Young and Marlon Byrd to help preserve the win.
"The biggest thing this afternoon was Kevin Jepsen; I don't think you could ask for anything more from a guy pitching in the back end of the bullpen than what he did," Scioscia said. "That was a great pitching performance from a guy with not a lot of experience."
Jepsen said there were times earlier this year when he was less effective because he threw too many balls and fell behind in counts. Now, he said, "when I come in I'm going to throw strikes and go after guys."
Speaking of great performances, Weaver scattered five hits through 7 1/3 innings for his 12th win of the year against three losses. He has won five consecutive decisions and nine of his last 10.
The right-hander also struck out a career-high 11 batters for the second consecutive game, and he drew a standing ovation from the 37,166 at Angel Stadium when he left the mound.
His only big mistake was giving up a two-run home run to Hank Blalock in the fifth inning.
"Just trying to get strike one" on hitters, Weaver said of his recent burst of strikeouts. "I kind of strayed away from that. My slider has been good and my changeup was good today.
"We were able to keep those guys off balance. The offense took care of the rest. Jepsen did a great job and that was awesome."
The victory enabled the Angels to widen their lead over the second-place Rangers to 4 1/2 games in the American League West. Texas has won eight of the teams' 11 meetings this season.
Weaver's outing "gives us a big lift, to have him come in and pitch this well against a team that's really been swinging the bats well against us," Scioscia said. "This is the way Weaver can pitch; he has that kind of stuff, that kind of command."
Aybar made a diving catch near second base for the second out of the ninth inning and helped closer Brian Fuentes on the way to his 31st save.
The shortstop also broke a 2-2 tie in the seventh inning when he hit a 1-and-0 fastball against reliever Eddie Guardado (1-2) into the left-field stands.
In the previous inning, Vladimir Guerrero had pulled the Angels even when he led off with a home run against starter Kevin Millwood.
Guerrero, who came off the disabled list Tuesday, is two home runs short of 400.
The Angels had taken a 1-0 lead in the fourth inning when Guerrero singled and advanced to second base on a wild pitch by Millwood. Kendry Morales then doubled over the head of center fielder Byrd to drive in Guerrero.
After the Rangers mustered only one hit against Weaver through the first four innings, Josh Hamilton doubled to lead off the fifth inning and then Blalock drove his home run to right field to give the Rangers a 2-1 lead.