Bunbury's second goal was intended to be a cross but it found the net in the 54th minute to break a 1-1 tie and the Revolution protected the lead at Soldier Field.
Midfielder Fabian Herbers scored the lone goal for Chicago. The Fire are just 1-5-1 over their past seven regular-season matches.
New England goalkeeper Matt Turner made a season-best six saves.
Chicago had a 17-9 edge in shots and placed six on target. The Revolution put just three on goal.
The sequence of the decisive tally began when Bunbury sent the ball in from the right flank. But it didn't curl back in toward his teammates and the ball drifted over the outstretched hands of Chicago goalkeeper Bobby Shuttleworth into the left corner of the net.
The brace allowed Bunbury to increase his goal total on the season to four.
Bunbury's first goal came in the third minute after the Fire failed to clear the ball out of their zone.
Midfielder Tommy McNamara eventually received the ball and he spotted Bunbury open to his right. He fed him with an accurate pass and Bunbury sent it through the legs of Bobby Shuttleworth for the goal.
The Fire knotted the score in the 22nd minute. A shot by defender Miguel Angel Navarro was wide to the right but directly toward Herbers, who knocked the short shot past Turner. The goal was Herbers' second of the season.
Shuttleworth's lone save came in the second minute of second-half stoppage time when he stopped a close-range shot by New England defender Alexander Buttner.
Chicago and New England played to a 1-1 tie at Foxborough, Mass., on March 7.