LA Galaxy used Targeted Allocation Money (TAM) to complete the transaction. As part of the deal, Barcelona have the option to buy back the Spaniard and will receive 50 per cent on any future sale.
The move comes on the same day the Galaxy transferred midfielder Rayan Raveloson to Ligue 1 side, AJ Auxerre, to open an international spot on their roster.
"We are pleased to welcome Riqui to the LA Galaxy," Galaxy head coach Greg Vanney said in a release. "Riqui is a very technical and well-educated player with incredible experiences for his age. He will fit seamlessly into our group and style of play."
Puig, 22, recorded two goals and three assists in 57 appearances (15 starts) in all matches with Barcelona.
"Riqui is a very talented player with a unique profile that brings something different to our squad," Galaxy technical director Jovan Kirovski said.