LA defeats Columbus, Colorado wins
Nathan Sturgis scored in the 47th minute and Deshorn Brown also scored for the Rapids, who have not lost at home to New York since 2007, winning four and drawing one since.
The Red Bulls would have moved above East leader Montreal with a victory but continued their disappointing recent form, winning two of their past six games.
Sturgis scored on a rebound goal in the opening minutes of the second half after the Rapids had missed several chances from close range.
Christopher Klute set up Brown with a perfectly timed cross in the 68th minute.
Keane beat Crew goalkeeper Andy Gruenebaum to the lower right corner in the 85th minute and again in the second minute of stoppage time to give the Galaxy just their third win in their past eight matches.
Bernardo Anor had put the Crew ahead in the 78th-minute, heading Federico Higuain's free kick from near midfield and looping the ball from the top of the box into the upper-right corner.
Gruenebaum gave away the first penalty for tripping Jose Villarreal, while Anor conceded the second for knocking down Keane as he awaited a cross.
Chivas extended its winless streak to 12, while Dallas has gone five straight without a victory.
Chivas outshot Dallas 17-6, holding a 6-1 advantage on shots on target, but has only scored 13 goals in 18 games.