The 10-man Red Bulls had broken a scoreless tie 1 minute into the extra time when Joel Lindpere took advantage of a botched New England clearing attempt and headed home a cross from Wilman Conde.
But only moments later, Barnes got the equalizer for the Revs (7-15-8), notching his first MLS goal off an assist by Ryan Guy on the last move of the game.
The draw kept New York (14-8-8) winless at Foxborough since 2002 -- a span of 16 matches.
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