The Goats had led 1-0 since early in the first half, but in the 88th minute Diego Fagundez sent a cross that confused goalkeeper Dan Kennedy and Barrett was able to get a foot on it, putting the ball past the far post.
The result gave a point to New England (5-5-6), which has gone three games without a victory.
Laurent Courtois tallied for Chivas (3-10-3), which snapped a seven-game losing streak.
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