The comeback spoiled a 13-strikeout effort from Dodgers starter Chad Billingsley, who tied a career high.
Dodgers reliever George Sherrill allowed Abreu's homer, his first of the season, to cut the Dodgers' lead to one.
Two batters later, Young hammered Jonathan Broxton's first pitch into the seats in left field to put Arizona in front in its final home game of the season.
Juan Gutierrez pitched the ninth for his 14th save.
Broxton (5-6) took the loss.
The Dodgers scored all of their runs with two outs. James Loney doubled in two runs to break a scoreless tie in the fifth inning and catcher Rod Barajas hit a two-run home run in the eighth for a 4-1 Dodgers lead.
The Diamondbacks came back in the bottom of the fifth with a leadoff triple from Miguel Montero, who scored on Cole Gillespie's sacrifice fly. But Billingsley struck out the next two batters to end the threat.
Billingsley allowed four hits in seven innings, escaping another jam in the seventh after another Montero leadoff triple.
The right-hander struck out two and got help from his defense with Barajas' short-hop scoop of a throw home to tag out Montero trying to score on a ground ball.
Ronald Belisario got the first out of the eighth for the Dodgers' bullpen, but Sherrill came on and walked Stephen Drew before Abreu's home run. Kelly Johnson singled, chasing Sherrill, and Young hit his home run off Broxton.
Notes: Dodgers outgoing manager Joe Torre doesn't appear to be interested in managing next season. Torre expressed that when asked about not managing in the postseason since 1996. "But you know, I think I've had enough of this stuff," the 70-year-old said Sunday. The Dodgers seem interested in keeping Torre in the organization in some capacity, Torre said, and he feels he still has the energy to do something. ... Diamondbacks interim manager Kirk Gibson, who hopes to return as manager for next season, reflected on his three months at the helm since being hired July 1. The Diamondbacks are 33-44 under Gibson and were 31-48 before he took over. "I've enjoyed the challenge. I look forward to an opportunity if I get it, at being better ... I get up, I get here early, I leave here late, make sure I take care of it, I'm very dedicated to it, and I want to try and make sure that I get smarter every day and help do my part." ... 3B Mark Reynolds missed a second straight game with a sore thumb. ... The top half of a broken bat during the game stuck upright in the infield dirt, drawing groans from the crowd. ... The probable starting pitchers for the next three Diamondbacks games, all at San Francisco, are right-handers Rodrigo Lopez, Ian Kennedy and Barry Enright.