Bradley Wright-Phillips recorded the hat trick for the Red Bulls (3-3-5), who suffered their first loss since April 12. Wright-Phillips has scored eight goals in his last four matches.
The Fire scored four goals in a span of 15 minutes in the second half. Shipp tallied two of them and assisted on another.
The match appeared to have the look of a rare high-scoring affair, with both sides scoring six minutes into the contest.
Forward Harry Shipp's first goal of the season, a chip shot from 18 yards, gave the Fire a brief 1-0 lead four minutes into match. Fellow forward Mike Magee appeared to be offside and Red Bulls coach Mike Petke argued strenuously to no avail.
The goal was Cahill's first of the season and the first allowed under 15 minutes by the Fire this season.
New York grabbed a 2-1 lead in the 39th minute on some superb playmaking from forward Thierry Henry.
The international superstar raced down the left flank past Fire defender Jhon Kennedy Hurtado and sent a cross to Wright-Phillips, who one-timed a shot into the goal from the right-corner of the box.
The goal was Wright-Phillips' sixth in the last four matches and the fourth assist for Henry this season.
Quincy Amarikwa slipped behind the Red Bull defense and received a long cross from midfielder Jeff Larentowicz that he promptly headed in for his team-leading fifth goal in the 49th minute, drawing the match even at 2-2.
Amarikwa, returned to the pitch after serving a one-match suspension for a red card he received against New England on April 19.
Shipp's second goal of the match, in the 54th minute, snapped a 2-2 tie. He dribbled around three New York defenders and fired a strike inside the penalty area. The shifty Shipp recorded the hat trick in the 58th minute to move the Fire ahead 4-2.
Wright-Phillips' ninth goal of the season, on a penalty kick in the 78th minute, drew New York to within a goal at 5-4. He also scored in the 67th minute.