Danks (5-9) lost his bid for a no-hitter when Billy Butler singled in the sixth. He ended up allowing two runs on four hits while fanning six and surrendering four walks.
Will Ohman finished up Chicago's victory with a scoreless ninth as the White Sox notched their eighth win in 10 games.
Lillibridge launched a three-run shot in the first inning, staking the Chisox to a 4-0 lead, while Carlos Quentin drove in a pair and Paul Konerko reached base five times.
Kansas City's Jeff Francis (4-13) could manage only 3 2/3 innings, allowing six runs on 10 hits.
Mitch Maier had a two-run single for the Royals, losers of eight of their last 10.
2014: NFL Cheerleaders [PHOTOS]
Julius Randle breaks leg during NBA debut