The Reds were trailing 2-1 in the sixth inning when they erupted for four runs to grab control of the contest.
Joey Votto walked and Phillips singled to lead off the frame against Cubs starter Travis Wood, and one out later Todd Frazier's ground single up the middle provided the equalizer, sending Phillips to third.
Then with Robinson batting, Phillips took off for home and Robinson laid down a sacrifice bunt to the right of the mound, putting Cincinnati ahead to stay.
Homer Bailey (3-3) earned the decision, yielding two runs on five hits over six innings. He walked three and struck out eight.
Wood (4-3) ended up allowing five runs on seven hits in a 5 2/3-inning performance.
Anthony Rizzo went 3-for-3 with an RBI for Chicago.