Noesi (1-4) ended a streak of 14 straight losing decisions dating to May 6, 2012 by scattering five hits and four walks over six innings of one-run ball.
Abreu went 2-for-4 with a home run and three RBI while Tyler Flowers added a solo homer for Chicago, which had dropped three of four coming into the game.
Dan Haren (5-4) gave up four runs on six hits in a six-inning start for LA, which went 0-for-7 with runners in scoring position and left a dozen on base.
The only run the Dodgers mustered came on Dee Gordon's bases-loaded sacrifice fly in the second inning.
Abreu belted a two-run homer in his return from the disabled list on Monday and duplicated the feat in the first inning Tuesday, as his shot down the left-field line was barely just out of Matt Kemp's reach and found the seats.
Abreu was up with runners on the corners in the third and pushed a run across with a productive fielder's choice grounder to third, and Flowers made it 4-1 in Chicago's favor with an opposite-field blast to right an inning later.
Gordon Beckham robbed Haren of a hit up with middle with a diving stop on a line drive to help leave runners on corners in the fourth, and Zach Putnam stranded two Dodgers in the seventh and eighth frames.