The Sox were up 5-3 after seven and just six outs away from Esteban Loaiza's 14th win, but a run allowed by Damaso Marte in the eighth put the Royals down one going into the ninth when Tom Gordon loaded the bases with one out.
After getting Ken Harvey to popout to first for the second out, Michael Tucker hit a soft groundball to the sliding Alomar at second, barely beating out the throw for an infield-single.
Lefty Scott Schoeneweis pitched the 10th and 11th innings, giving up one run, to pick up his first win with the Sox.
Loaiza didn't have his best stuff, allowing two early runs in the third and another in the fifth, but battled for seven innings, striking out seven.
The first three runs for the Sox came on two homers and a sacrifice fly by Carlos Lee. The homers were his 21st and 22nd on the season. Lee went 3-5 with the homers, a single and a stolen base.
Konerko put the Sox ahead 5-3 in the sixth with a two-run homer, scoring Carl Everett, who singled. Konerko went 2-5 and is 24-65 with six homers and 22 RBIs since July 11.
The White Sox will now head to Seattle to play a three-game series starting Friday at 9:05 p.m. CT, with Bartolo Colon going up against Freddy Garcia. The game will be televised on Fox Sports Net.