Twins starter Anthony Swarzak (3-5) lasted 4 1/3 innings, allowing eight runs -- six earned -- on six hits.
He entered the fifth with a 6-3 lead, but Paul Konerko ripped an RBI double before the tying runs scored on Michael Cuddyer's throwing error. Burnett entered and issued a walk to load the bases, then hit Beckham two batters later.
Juan Pierre added an RBI single and Alejandro De Aza belted a three-run homer for the White Sox, who have won five straight and seven of nine. They remain five games behind the AL Central-leading Tigers, who beat the Royals in extra innings earlier Tuesday.
Rookie starter Zach Stewart gave up six runs in 4 2/3 innings for Chicago, but Will Ohman (1-3) recorded the final out in the top of the fifth to earn the win.
Sergio Santos fanned three in a scoreless ninth to record his 28th save of the season.