Springer connected on the first pitch of the day as the AL West-leading Astros won for the ninth time in 10 games. He also threw out a runner at the plate from right field.
Greinke (12-4) allowed two runs on seven hits in winning for the second time since Houston acquired him from Arizona in a deal at the trade deadline. The 35-year-old righty struck out six, walked two and hit a batter in a 102-pitch outing that was sharper than his Astros debut last week when he allowed five runs in six innings to Colorado.
Will Harris, Ryan Pressly and Collin McHugh each followed Greinke with a perfect inning.
Robinson Chirinos drove in two runs late with a pair of singles.
Eloy Jimenez and Jose Abreu drove in runs with doubles as Chicago came back to tie it from an early 2-0 deficit.
The Astros took the lead for good in the sixth when Yordan Alvarez scored an unearned run on the first of three passed balls charged to Welington Castillo.
Alvarez raced home from third to break a 2-all tie when Castillo couldn't handle Dylan Cease's shoulder high pitch to Aledmys Diaz. Castillo recovered the ball and threw to Cease covering the plate, but Alvarez narrowly beat the tag on a play that withstood video review.
Jake Marisnick scored from third in the seventh to make it 4-2 when Castillo failed to catch a high pitch to Alvarez.
Making his seventh start, Cease (2-5) allowed four runs -- two earned -- in six-plus innings. The rookie right-hander yielded five hits and walked five.
Springer launched Cease's first pitch to the back of the left-field seats for his 26th homer and 33rd career leadoff shot.
Greinke allowed three singles in the second, but escaped unscathed when Springer threw out Castillo at the plate to end the inning.
Altuve led off the third with a line shot just over the left-center wall to give Houston a 2-0 lead.
Jimenez's double to the left-center gap in the fourth drove in Abreu from first to cut it to 2-1. Abreu's RBI double with two outs in the fifth tied it at 2.
With Houston ahead 4-2, Chirinos drove in insurance runs in the eighth and ninth.
Tuesday's straight doubleheader was scheduled after Monday night's game was postponed by rain.
Astros: RHP Brad Peacock (right shoulder discomfort) was in Houston's clubhouse following two rehab appearances for Triple-A Round Rock and said he felt "really good." Manager A.J. Hinch said Peacock will return to the minors for one or two more outings, starting on Thursday, before rejoining the Astros as a reliever. Peacock was 6-6 with a 4.13 ERA in 17 games (15 starts) before going on the IL on June 28. . RHP Josh James (sore right shoulder) threw 20 pitches in Florida on Sunday. Hinch said the next step will be to intensify James' rehab program.
White Sox: INF/OF Leury Garcia (bruised finger on right hand) sat out the first game. . Manager Rick Renteria said 3B Yoan Moncada (strained right hamstring) is progressing as he work outs with the team, but there's no timetable for his return.
UP NEXT: Astros RHP Gerrit Cole (14-5, 2.87 ERA) starts the second game of the doubleheader against White Sox RHP Ivan Nova (7-9, 4.80). Cole seeks to extend his career-best winning streak to 11 games; the last team to beat him was the White Sox on May 22. Nova is 3-0 with a 0.64 ERA in his last four starts.
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Astros catcher takes a huge hit but gets the out at home
Robinson Chirinos takes on the runner at home plate and holds on to the ball for the out.