Axelrod (2-3) limited the Red Sox to two runs on four hits over six innings, walking one and striking out two en route to his second victory in two starts.
Addison Reed tossed a scoreless ninth inning for his 15th save as the White Sox broke a two-game skid.
Boston starter Jon Lester (6-1) absorbed his first loss of the year in his 10th start, yielding six runs (five earned) on seven hits over six full innings.
Jarrod Saltalamacchia went 2-for-3 with a two-run homer and Will Middlebrooks also drove in pair for the Red Sox, whose five-game winning streak was snapped.
Notable deaths of 2014 [PHOTOS]