The 23-year-old Johnson (1-2), making his third big-league appearance since being called up this month, scattered four hits over six scoreless innings. The young right-hander walked two and struck out eight.
Alexei Ramirez led Chicago at the plate, going 4-for-4 with a solo home run and three RBI.
Jordan Danks belted a two-run homer while Adam Dunn and Dayan Viciedo each contributed a solo shot for the White Sox, who banged out 13 hits.
Twins starter Liam Hendriks (1-3) didn't make it out of the first inning, allowing seven runs on five hits and three walks.
Chris Parmalee got the only RBI for Minnesota, which dropped the opener of a three-game series.