The situation for Konerko came with the help of a ninth-inning throwing error by Indians' closer Danys Baez, allowing the tying runs to score.
The error came when Roberto Alomar, up with two outs and men on second and third, hit a high comebacker to Baez, who rushed the play, causing a bad throw down the right field line. Scoring was Carl Everett and Joe Crede, who reached on an infield-single and walk, in order.
White Sox starter Mark Buehrle continued his recent streak of impressive starts in giving up four runs (two earned) in 7 1/3 innings. All of the earned runs came in the first inning on singles by Coco Crisp and Casey Blake, and a two-run double by Milton Bradley.
That put the Indians on the board first with a 2-0 lead, a lead that was expanded to 3-0 when a throwing error by Roberto Alomar on a double play attempt put men at the corners for Ben Broussard. Broussard hit a RBI-sacrifice fly.
The Sox cut the deficit to 3-2 with a run in the sixth and eighth innings, both on hits by Carlos Lee. Lee went 3-5 with two RBIs.
Another gift run was given to the Indians in the eighth when a throw to second by Miguel Olivo on a steal attempt by Jody Gerut went off the glove of Roberto Alomar. The error allowed Milton Bradley, who was stealing third on the play, to score. Damaso Marte was on the mound, but the run was charged to Buehrle.
Marte threw 1 2/3 scoreless innings, enabling the White Sox to come back with two runs in the ninth.
Konerko's homer was setup by a Magglio Ordonez leadoff double and Lee single.
Tom Gordon pitched a scoreless ninth to pick up his second save of the season.
Game two of the double-header and three of the series will be played tonight at 6:05 CT, with Mike Porzio going up against Jake Westbrook. The game will be televised on WGN.