Dallas (2-0-1) sits atop the Western Conference with an MLS-best seven points.
Castillo opened the scoring in the 71st minute off a Mauro Diaz set piece, and Watson outran defender Tony Lochhead on a long ball and then juked goalkeeper Ian Kennedy for the eventual game-winner in the 78th minute.
''We have a good team,'' Watson said. ''The guys are running really hard. To get three goals so fast was like a reward because we had them on the back foot from the get-go of the game. I think the guys came out in the second half and decided we were going to go for the kill.''
Chivas forward Erick Torres grabbed his third goal of the season in the 81st minute to put the pressure back on the hosts, but Michel, a second-half substitute, slotted in a low, left-footed free kick in the 86th and Chivas (1-1-1) lost for the first time this season.
Each team had 10 shots, but Dallas had nearly 60 percent of the possession.
''When you make those mistakes on set pieces and concentration, you don't deserve (points),'' Chivas coach Wilmer Cabrera said. ''You have to get punished in order for us to improve because we've been working - set pieces - we've been working on that, and today we lost concentration and then we got punished.''
Dallas midfielder Andrew Jacobson left the game in the 67th minute with what coach Oscar Pareja said was an ''extreme'' left ankle injury. Jacobson was on crutches after the game in the Dallas locker room.
Jacobson, who came on in the 11th minute after Andrew Moffat tweaked a muscle in his left leg, hurt his ankle going for the ball and getting tangled with a Chivas player near midfield.
Chivas' Oswaldo Minda was replaced in first-half stoppage time with a right-leg injury that will be evaluated when the team gets home.
Castillo's strike came from a set piece that Diaz snapped into the box from 36 yards out. Veteran Honduran midfielder Hendry Thomas received the ball and headed it toward Castillo about 6 yards out, and Castillo swept it off Kennedy's outstretched hand for the opener.
Kennedy kept Chivas in the game in the first half when he saved a 30th-minute, point-blank header from Blas Perez, pushing the shot off the line to keep the game square.
Both teams then started fouling more aggressively in the midfield and four yellow cards were shown in the final 15 minutes of the half, including three to Chivas players. The visitors picked up two more yellow cards in the second half, and Thomas earned an ejection during stoppage time.