Eric Avila put the visitors ahead in the 55th minute, but Bernier converted after Mario de Luna had fouled Sanna Nyassi to ensure the Eastern Conference leaders took a point.
They saw their lead cut to one point, though, as Sporting Kansas City claimed a 2-1 win at the Chicago Fire.
Benny Feilhaber, starting for the first time in seven games, volleyed the opener in the sixth minute and Graham Zusi added a second two minutes later.
Mike Magee pulled one back in the 38th minute for the Fire, who had Daniel Paladini sent off in stoppage time.
The Los Angeles Galaxy continued their rise up the Western Conference with a 2-0 win over FC Dallas.
Marcelo Sarvas headed home Robbie Keane's cross to open the scoring in the 65th minute and Hector Jimenez latched on to Robbie Rogers' assist to add a second 10 minutes from time.
Bernardo Anor headed home a corner in the fourth minute and the Columbus Crew shut out the 10-man Portland Timbers for a 1-0 win.
Pa-Modou Kah was sent off after 11 minutes for a foul on Aaron Schoenfeld in the penalty area, but Federico Higuain missed the spot-kick.
The day's other game between the Colorado Rapids and DC United, finished goalless.