Houston (3-4-2) snapped a six-match winless skid and equaled its offensive output from the last seven games. Brad Davis and Will Bruin also scored for Houston.
Erick Torres scored the lone goal for Chivas USA (1-5-3), which had its winless streak extend to eight.
Chivas finished with 10 players as goalkeeper Tim Melia was sent off in the 67th minute for tripping Will Bruin inside the penalty area.
Chivas opened the scoring in the 7th minute when Torres converted a penalty kick, but Houston scored the next four goals. Chivas goalkeeper Tim Melia was handed a red card for tripping Bruin inside the penalty area. Barnes converted the penalty in the 69th minute to provide the final margin.