Couch, with Double-A Portland for most of the season prior to Saturday, earned the win for the Sox, shutting out the Bulls over 6 2/3 innings.
Castillo, the leadoff hitter, finished 2-for-4 with a pair of doubles and two RBIs. Lavarnway also collected a pair of hits, including a solo home run. His performance helped earn him the Governors' Cup MVP.
"Regardless of the level, if you're the last team standing, there's significance to it," Red Sox manager John Farrell said.
Miguel Celestino cleaned up the final 2 1/3 innings for the save.
The Red Sox move on to face the winner of the Pacific Coast League final (Omaha or Reno) for Triple-A bragging rights.
"In the message exchanged with [Pawtucket manager Kevin Boles] last night, it was clear excitement on his part, and to walk away after a long season with a lot of work done, congratulations to them," Farrell.
Castillo and four other Pawtucket players yet to be named will be joining Boston on Wednesday in Pittsburgh.
Jackson Alexander is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.