The 38-year-old third baseman, featuring for the first time this year following a lengthy recovery from a pre-season hip injury, ended the night 1-for-4 with a single, two fly-outs and a strikeout at US Cellular Field.
Rodriguez was earlier on Monday handed a ban - starting on Thursday and covering the remainder of this campaign and all of next - following an investigation into the now defunct Biogenesis clinic in Miami. He swiftly announced his intention to appeal and "fight this through the process".
Alex Rios drove in four runs and Alexei Ramirez added four hits as the White Sox snapped a 10-game losing streak.
In other American League action on Monday, the Central-leading Detroit Tigers extended their winning run to nine games with a 4-2 come-from-behind success against the Indians in Cleveland, Alex Avila proving the hero with a three-run homer in the ninth inning.
Brett Oberholtzer allowed four hits over seven shutout innings as the Houston Astros overcame the East-leading Boston Red Sox 2-0, while a career-high five RBIs from Eric Hosmer spurred the Kansas City Royals thrash the Minnesota Twins 13-0.
Elsewhere, the Texas Rangers overcame the Los Angeles Angels 5-2 and the Toronto Blue Jays were 3-1 winners against the Seattle Mariners.
In the National League, the East-leading Atlanta Braves edged the second-placed Washington Nationals 3-2, Justin Upton smashing the go-ahead homer in the eighth inning to secure Atlanta's 11th straight win.
Zack Greinke's game-winning RBI single secured his 100th career win as the Los Angeles Dodgers clinched their 15th road win in a row - 3-2 against the St Louis Cardinals - while the San Francisco Giants toppled the Milwaukee Brewers 4-2.