McCutchen, who went 2 for 5, recorded the 1,00th hit of his career with an eighth-inning single.
Starling Marte and Jung Ho Kang had two RBIs each for the Pirates, with Jordy Mercer driving in a run.
Cole (4-0) allowed an unearned run in six innings, striking out eight and lowering his ERA to 1.76. The right-hander is 6-0 in six career starts against the Cubs.
Miguel Montero went 3 for 4 and drove in Chicago’s lone run.
Chicago’s Kyle Hendricks (0-1) allowed two runs and five hits in five innings.