Working his second inning of relief, Brian Sweeney (1-2) retired his first batter in the 14th before Abreu drove the next pitch to left-center for his 18th homer.
Rich Thompson (1-0), Los Angeles' fifth reliever, tossed two perfect innings to get the victory.
It was the fifth blown save opportunity in 13 chances this season for Rodney, who inherited the closer's job after Brian Fuentes was traded to Minnesota on Aug. 27. Two nights earlier against Cleveland, Rodney gave up a tying RBI single to pinch-hitter Jayson Nix in the ninth before the Angels won it in the 16th.
The 30 innings in consecutive games tied an Angels franchise record, matching the mark set April 13-14, 1982, in Anaheim, when they beat the Mariners 4-3 in 20 and 2-1 in 10 to begin a seven-game winning streak.
Former Angels third baseman Chone Figgins, who shifted to second after signing a $36 million, four-year contract with Seattle in the offseason, made a spectacular play in the 10th to keep the game going.
Mike Napoli hit a grounder up the middle with runners at first and second, but Figgins backhanded the ball behind the bag and made a no-look backhanded flip with his glove to shortstop Josh Wilson, who completed the inning-ending double play.
Angels pinch-hitter Juan Rivera grounded into another inning-ending double play in the 12th with runners at first and second.
Weaver allowed an unearned run and five hits while lowering his ERA to 3.02, but remained winless in six starts since beating Detroit on Aug. 6. He struck out six to raise his career-high total to 211 and regain the major league lead from Seattle's Felix Hernandez, who starts Saturday night against Ervin Santana.
David Pauley gave up three runs and five hits over 6 2-3 innings for Seattle in his 16th big league start. This was the first time in Pauley's last four starts that his teammates scored a run while he was in the game.
But the 27-year-old rookie has nothing on Weaver, whose teammates managed only one run over his previous four outings — an eighth-inning solo homer by Torii Hunter last Saturday at Oakland. The Angels had totaled only 26 runs in Weaver's previous 14 starts after beating the Cubs 12-0 at Wrigley Field on June 19.
Seattle's Ichiro Suzuki extended his hitting streak to 11 games with a bunt single in the third and continued to second on catcher Jeff Mathis' throwing error. It was Suzuki's 2,214th big league hit, tying Joe DiMaggio for 167th place on the career list.
Suzuki took over the AL lead in hits by one over Texas Rangers outfielder Josh Hamilton. The Japanese star has led the majors in hits six times during his nine previous big league seasons and each of the past four.
The Angels took a 2-0 lead in the fourth on Howie Kendrick's sacrifice fly and an RBI single by Mathis. Seattle got one of the runs back in the fifth when Kotchman singled, took third on Michael Saunders' single and continued home as Hunter's one-hop throw from right field skipped past third baseman Alberto Callaspo.
The error was only the fourth by Hunter in his last 407 games in the outfield, and the second for the nine-time Gold Glove winner since shifting from center field to right when Peter Bourjos was promoted from the minors on Aug. 3.
Mathis, who fouled out on a squeeze attempt Wednesday night before getting a game-ending sacrifice fly in the 16th, laid down a perfect bunt in the sixth as Erick Aybar scooted home with the Angels' third run.
NOTES: Weaver has left with the lead in five of his eight no-decisions. ... Most of the Fireworks Night crowd of 44,203 had long departed by the time it was announced at the end of the 13th that the pyrotechnics would be postponed until Saturday. ... Hunter's error, which followed two by Mathis, increased the Angels' season total to 100. ... The Mariners' offense reached the 1,000-strikeout mark for the third straight season and 11th time in the franchise's 34-year history when Weaver fanned Adam Moore in the seventh inning.