Zambrano (8-6) allowed seven hits, walked three and struck out five for the Cubs.
Chicago broke the game open with three runs in the sixth. Starlin Castro and Jeff Baker led off with singles and Soto's double to left drove in Castro for 2-1 advantage.
Alfonso Soriano and Carlos Pena added RBI in the Cubs' big inning.
Lee, playing his first game for Pittsburgh since being acquired from Baltimore Saturday, blasted a pair of homers but it wasn't enough for the Bucs, who lost their fourth straight.
Paul Maholm (6-11) absorbed the loss, surrendering four runs on six hits over six frames.
Swim Week Miami Beach 2014 [PHOTOS]