Luka Stojanovic had one goal and one assist for Chicago, which remains winless (0-6-1) on the road this season. The match was the first of a six-game road trip for the Fire (5-10-5, 20 points).
Francisco Calvo also scored for the Fire, who have been outscored 17-5 on the road this season. Chicago had a four-game overall unbeaten stretch (2-0-2) halted.
Higuain was a key figure in Miami's winning goal as he found Victor Ulloa, who then fed Pizarro the ball. Pizzaro sent a hard right-foot shot to the left of Fire goalkeeper Bobby Shuttleworth into the right corner of the goal.
Miami had a 12-8 edge in shots and placed four on target to Chicago's three.
Shuttleworth and Miami counterpart Nick Marsman had each made one save.
Chicago took a 2-1 lead just three minutes into the second half. Stanislav Ivanov threaded a pass to a wide-open Stojanovic, who sent a straight-on right-footed shot into the upper left corner of the net.
Miami tied the game in the 62nd minute as Higuain delivered a crisp pass to Robinson. The No. 1 overall pick in 2020 then sent a right-footed shot into the top right corner of the net.
Higuain had a chance to put Miami ahead in the 87th minute but sent a right-footed blast over the net.
Miami struck first in the 34th minute as Robinson took a right-footed shot that was saved by Shuttleworth. But Vassilev was there for the rebound and booted a right-footer into the net.
The Fire knotted the score six minutes later when Stojanovic took a corner kick and sent the ball into the box. Calvo sent a glancing header toward the net that caromed off the hands of Marsman and into the net.