With the Impact trailing 1-0 in the 80th minute, Chivas defender Mario de Luna brought down Sanna Nyassi with a sliding tackle, setting up a penalty try.
Bernier made a fake, sending goalkeeper Dan Kennedy to his left, then calmly booted it to the right for the equalizer.
The draw enabled the Impact (9-4-4) to retain a one-point lead atop the Eastern Conference with 31 points.
Eric Avila tallied in the 55th minute for the Goats (3-10-5), who extended their winless streak to 13 games.
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