After Fuentes retired David Murphy on a liner to first for his 31st save, Weaver moved to 12-3, and the Angels restored their 4 1/2-game lead over the resilient Rangers in the AL West race.
"That was a great win for us," Weaver said.
The drama peaked in the top of the eighth, after Vladimir Guerrero's sixth-inning homer against Kevin Millwood had tied it and Aybar's shot had given the Angels the lead.
Weaver departed to an ovation after a walk, single and bunt had runners on second and third with one out.
In came Jepsen, the gas-pumping right-hander from Nevada who turned 25 two weeks ago.
"Nothing but four-seamers and cutters," Jepsen would say later.
Facing Michael Young, one of the game's most feared clutch hitters, Jepsen jumped ahead in the count with a cut fastball, but Young worked the count full before Jepsen sent a 97-mph heater past the Rangers' All-Star third baseman.
"You just don't see Michael Young not putting the ball in play in a situation like that," Angels manager Mike Scioscia said.
Jepsen was not done. Marlon Byrd stepped up, having doubled in three at-bats against Weaver. Jepsen jumped ahead 0-2 with a pair of 97-mph fastballs and then left one in a location that Byrd managed to hit, but not hard, to right fielder Bobby Abreu.
"Usually I'm upset when I don't get the job done, but Jepsen just stuffed the ball down my throat," Byrd said. "He made some unbelievable pitches."
Jepsen used the crowd and the game's magnitude to ride an emotional wave.
"I came in with the attitude that I didn't want anybody to hit the ball," he said. "My first thought was to get ahead, which I did, and then make good pitches. I had so much adrenaline, I didn't have to worry about velocity -- it was going to be there.
"I had good life when I struck out Young. On Byrd, I was climbing the ladder, and he hit an 0-2 four-seamer that was in a good location."
Jepsen got an official hold, but this clearly was a save.
"My whole mindset," Jepsen said, "is that I don't want to let another pitcher's baserunners score. In any situation, that's how I feel, but especially when it's a guy who's having a game like Jered had. He was great."
Weaver produced 11 punch-outs with 69 of his 100 deliveries in the strike zone. Coming on the heels of 11 Ks in 6 1/3 frames in a win at Minnesota on Sunday, it was the first time he'd reached double-figure strikeouts in consecutive games.
"Weaver does what he always does -- he paints the corners with every pitch," Byrd said.
Millwood, the Rangers' ace, held up his end of an old-fashioned duel.
"Every time I've faced Texas, we've had some great battles," Weaver said. "He's a great pitcher. We were able to get one more.
"Those guys have had our number this year. Obviously, I wanted to get deep in the game. I did that. Jepsen came in and did a great job, and Tito shut it down."
It was a two-game swing in the standings, with John Lackey engaging young Derek Holland in Sunday's series finale.
The Rangers have taken eight of 11 in the season series.
Bringing his team even at 2, Guerrero unloaded on a 1-0 Millwood slider, driving it into the seats in the left-field corner to lead off the sixth. It was his sixth homer of the season and second in five games since coming off the disabled list.
"Not many guys can go down and hit that pitch," Scioscia said.
Guerrero had singled with one out in the fourth and scored on Kendry Morales' two-out double after a Millwood wild pitch. This handed Weaver a lead that lasted all of two pitches.
Josh Hamilton raked a fastball for a double off the wall in right center leading off the fifth, and Hank Blalock followed with a booming homer to right on a Weaver slider.
Weaver went right back to work, striking out the next two men he faced and five of the next seven.
Millwood frustrated the Angels, who stranded seven runners from the third through the sixth frames.
But Guardado, leaving a fastball in the middle to Aybar, fell to 1-2 with the shortstop's fifth homer, and he showed his astonishing quickness two innings later in stealing a hit from Hamilton.
"The two, both of them," Aybar said, grinning, when asked which felt better, the homer or the superlative defensive play.
"That's what that kid can do," a satisfied and drained Scioscia said. He was referring to Aybar, but the words on Saturday applied to Weaver and Jepsen as well.