"We've had to manufacture some runs, but when guys can hit the ball out of the park and score some runs that way, it's always nice as well," said Crosby, who snapped an 0-for-15 slump, including a 12 at-bat hitless streak on the road, with his second homer of the season.
Outman (3-0) allowed seven hits and three runs -- two earned -- over 6 2/3 innings. Andrew Bailey got five outs for his third save in six chances.
Once again the White Sox couldn't solve a rookie pitcher they had never seen previously. Chicago is now 1-6 this season when facing a pitcher for the first time and those hurlers have a combined 1.70 ERA against the White Sox in nine starts.
And they'll see their fourth straight Oakland rookie on Thursday with Brett Anderson scheduled to pitch. Chicago was shut down by Vin Mazzaro in his major league debut Tuesday night.
"We lost against another rookie," Chicago manager Ozzie Guillen said. "Quote me on this: The Oakland A's are going to have a tremendous pitching staff the next two or three years. The two or three guys we faced, they are good and the know what they are doing out there."
Chicago starter Clayton Richard (2-1) gave up seven hits and five runs and surrendered all three of Oakland's homers.
"He just outpitched me and that was kind of the story tonight," Richard said of Outman.
Crosby and Powell hit Richard's first two pitches of the fourth inning over the fence in left center to give the A's a 4-2 lead.
Giambi hit a two-out homer in the fifth, his seventh of the season.
"It was just a matter of time before the offense was going to strike again and they hit the ball hard tonight so it was just a great confidence booster," Outman said.
But Richard struggled in the top of the second, walking two and then giving up consecutive singles to Powell, Aaron Cunningham and Rajai Davis -- the bottom three in the batting order, all hitting under .200 -- to tie the game. The A's still had bases loaded and one out before Orlando Cabrera's grounder to third forced Powell at the plate and Adam Kennedy rolled out to first.
Chicago sliced it to 4-3 in the fourth on singles by Paul Konerko and Josh Fields and a throwing error on second baseman Kennedy as he tried to make a relay to first to turn a double play after Brian Anderson's grounder.
But Giambi's homer restored the two-run lead in the fifth.
The White Sox said they will purchase the contract of infielder Gordon Beckham from Triple-A Charlotte before tomorrow's game against Oakland. The 22-year-old Beckham was the No. 8 overall selection in last year's draft out of Georgia. Infielder Wilson Betemit will be designated for assignment. ... DeWayne Wise replaced Anderson in center field to start the fifth. ... Opposing pitchers are 6-1 this season when facing the White Sox for the first time in their careers. Oakland's Vin Mazzaro beat the White Sox in his major league debut Tuesday night. ... Many fans were bundled up with temperatures in the low 50s and a chilly wind.