The doubleheader was scheduled after the Athletics and Astros decided not to play Friday night in order to raise awareness of racial inequality in the wake of the police shooting of 29-year-old Jacob Blake, a Black man, in Wisconsin.
Tucker put Houston ahead to stay with his sixth homer in the first inning, a drive off Chris Bassitt (2-2) that went just inside the right-field foul pole.
Tucker is batting .365 (19 for 52) with five homers and 17 RBIs in his last 16 games.
''He's actually carrying us right now,'' Houston manager Dusty Baker said. ''I'm glad he is. He's coming into a zone. He's playing with a lot of confidence. He's driving in some big runs.''
Reddick led off the fourth with his second homer.
McCullers (3-2) allowed two runs - one earned - and six hits with seven strikeouts and one walk.
''I was really happy with the changeup today,'' McCullers said. ''And my breaking ball and really most of my stuff. I made a little adjustment midweek. I threw two bullpens this week with the hurricane days, and I made a little adjustment with my front side. It felt like it was coming out more consistent on that two-seam line.''
McCullers has allowed two or fewer runs in three of his last four starts.
''I had gotten away from who I am, especially on the road starts,'' McCullers said. ''I have to remember I am a stuff guy. ... I just kind of went out there and was really confident in my ability to throw strikes on my pitches.''
Matt Chapman drove in each of Oakland's runs. Bassitt allowed six hits and walked three in three innings.
''Just a little off with his command, getting behind some, it's two out, nobody on and all of a sudden, it's 3-0 after that so that gets you on the run a little bit,'' Oakland manager Bob Melvin said. ''In a short game, we couldn't give them too much rope.''
Ryan Pressly pitched the seventh for his fifth save.
The teams were originally scheduled to celebrate Jackie Robinson Day on Friday, but celebrated it during the first game of the doubleheader instead, with every player and coach wearing No. 42.
DOUBLEHEADERS IN HOUSTON
Saturday's doubleheader was the second for the Astros this week and third in Minute Maid Park history, with the other one on Sept. 2, 2017, against the Mets following Hurricane Harvey. The Astros and Angels moved up Thursday's game to Tuesday ahead of Hurricane Laura, which made landfall in Louisiana on Thursday morning. Houston's game on Wednesday against the Angels was postponed.
Astros: OF George Springer returned to the lineup after missing Tuesday's doubleheader with a bruised left elbow. ... INF Aledmys Diaz was reinstated from the 10-day injured list.
Athletics: RHP Frankie Montas (2-2, 5.22 ERA) will start the second game. Montas has allowed 13 runs in 6 1/3 innings over his last two starts.
Astros: RHP Zack Greinke (1-0, 2.20 ERA) will start the second game of the doubleheader. Greinke has gotten no-decisions in five of his six starts this season.