Both sides were unbeaten through two matches, with a win and a draw apiece, and a scoreless stalemate was on the cards as it remained 0-0 through 70 minutes at Toyota Stadium in Texas.
But Dallas would open a four-goal frenzy in the final 20 minutes. Fabian Castillo got on the end of Hendry Thomas' header from a set piece, and Chivas goalkeeper Dan Kennedy was unable to keep out the close-range shot.
The hosts' lead was doubled on 78 minutes, when Je-Vaughn Watson capitalised on a defensive blunder from Chivas' Tony Lochhead to round the keeper and tap into the empty net.
Chivas gave themselves a lifeline three minutes later, as Erick Torres cleverly flicked in a goal from a low cross, but Dallas would seal the three points shortly after.
Michel powered home a free-kick from range that deflected off the Chivas wall and into the back of the net.
Thomas' good work in assisting Castillo earlier was long forgotten, though, when he was sent off for a two-footed lunge on Adolfo Bautista in injury time.
Vancouver Whitecaps were held to a 0-0 draw on the road at New England, who picked up their first point of the season.
Whitecaps sit second in the West, ahead of fellow unbeaten sides Real Salt Lake and Colorado Rapids.
Salt Lake were held 1-1 at home to Los Angeles Galaxy, with Alvaro Saborio's opener cancelled out by Robbie Keane's 35th-minute goal, while the Rapids trumped the winless Portland Timbers 2-0.
Jermain Defoe made it three goals in two games for Toronto FC, as the former Tottenham man fired the Canadian outfit to a 1-0 home win over DC United.
Toronto are one of three teams with six points from two matches in the Eastern Conference, with each of Columbus Crew and Houston Dynamo also carrying perfect records.
Columbus' Bernardo Anor scored a brace in their 2-1 win at home to Philadelphia Union.
Sporting Kansas City earned their first win in their title defence, beating San Jose Earthquakes 1-0 courtesy of a Dominic Dwyer penalty.
