Johnson said the trio heard that they were due to be traded Wednesday and had it confirmed Thursday.
Wood is one of two starting pitchers, along with Miami's Mat Latos, that the Dodgers added to their rotation. The 24-year-old is 7-6 with a 3.54 ERA this season for Atlanta and will make his Dodgers debut Tuesday.
"I was hoping this would be the final stop for me when it was all said and done over the past 48 hours," Wood said. "If there was a team I would ever want to play for, it's here. My first road appearance in the big leagues was here and playing them in the playoffs in 2013. I just love the atmosphere and the fans."
Johnson and Avilan will plug holes in Los Angeles' bullpen that is 23rd in the Majors with a 3.87 ERA as a unit. The pair aren't exactly fresh arms, however, as Avilan is fourth in the Majors in appearances with 50, while Johnson is sixth with 49 appearances going into Friday's game.
"Every team that wants to win in October, that's the key factor. Getting those big outs at the end of the game," Johnson said. "That's why we're here."
Johnson is a former closer for the Orioles in 2012 and 2013 where he converted 101 saves in the two-year span. He has nine saves this season while Jason Grilli was injured earlier this season but figures to work as a setup man to current Dodgers closer Kenley Jansen. Johnson is sixth in the Majors with 20 holds.
Avilan has a 3.58 ERA this season and will join J.P. Howell as another left-hander out of the Dodgers' bullpen. He has 11 holds on the season.
"We're going to be in the game in key situations and that's what every reliever strives for," Johnson said. "To be in the game just to be in the game and to be in the game in a good situation is totally different. That's what this opportunity affords to us."
Steve Bourbon is an associate reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.