David Foley and Josh Hansen also scored for the Islanders, a member of the United Soccer League's first division. The Galaxy scored on an own goal by Puerto Rico defender Richard Martinez.
Islanders goalkeeper Bill Gaudette, who played for the Columbus Crew during the 2005-06 season, made seven saves.
The winner of the two-game series, determined by overall goals scored, will advance to the group stage. The second game will take place August 4 in San Juan.
Foley gave Puerto Rico a 1-0 lead in the 26th minute. Galaxy defender A.J. DeLaGarza headed David Horst's 14-yard header off Hansen's corner kick away from the net, but Foley converted the loose ball from 2 yards.
Horst kept Los Angeles from scoring the tying goal in the 31st minute by clearing Sean Franklin's 8-yard shot away from an open net.
Addlery made it 2-0 in the 45th minute with an 8-yard shot off Galaxy goalkeeper Josh Saunders' hand.
Hansen's 29-yard line drive in the 56th minute extended the lead to 3-0. Addlery scored his second goal in the 81st minute on a 19-yard line drive for a 4-0 advantage.
Martinez deflected Edson Buddle's shot into his own net in the 83rd minute.