The Canadian side were dealt a blow in the first half when former Aston Villa striker Luke Moore was harshly sent off after his leading elbow made contact with Chris Ritter's head as they both went for a header in the 28th minute.
Despite that setback, the visitors took the lead three minutes from the half-time break when Jermaine Defoe's cross from the right picked out an unmarked Jackson and he headed past Fire goalkeeper Sean Johnson.
The home side were level 12 minutes after the restart when a well-worked short corner saw Harrison Shipp in space in the penalty area and his deflected shot found its way past Joe Bendik at the near post.
Both sides had other chances to claim victory but neither could find a winning goal.
In the day's other game, the San Jose Earthquakes replaced Chivas USA at the bottom of the Western Conference after falling to a 1-0 defeat to Wilmer Cabrera's side.
Chivas claimed all three points courtesy of a Erick Torres strike eight minutes into the second half.