Dunn, who hit just 11 home runs last year, leads the majors with 35 this season.
Saturday's homer went 428 feet to left-center, but Dunn found it difficult to enjoy his accomplishment in the wake of a 9-4 loss to the Kansas City Royals.
"It's just about the worst-case scenario, getting it like this," Dunn told reporters after the loss. "We've played two bad games. Tonight was a disaster from the get-go. Obviously, I would have wanted it to come in a win."
Chicago trailed 5-2 before Dunn drove a 2-2 pitch from Tim Collins over the wall in the eighth.
The homer was the fourth this week for Dunn, who hit two solo shots on Monday against the Toronto Blue Jays and one more on Wednesday. He now ranks 50th on baseball's all-time home run list.
Konerko hit his milestone homer on April 25 against the Oakland Athletics.
The Associated Press contributed to this report.