The 22-year-old forward has signed a four-year deal at Goodison Park and is expected to be part of a South American double swoop for boss Roberto Martinez, who also hopes to tie up a deal for Argentinian central defender Ramiro Funes Mori soon.
Rodriguez scored 19 times in 71 games for his hometown club and Martinez hopes he can mature on Merseyside.
“Leandro is one of those young players that we have identified to give them a path and allow them to develop in our system,” the Spaniard said.
“We’re very excited; he’s someone who can produce a goal out of nothing, is hard working with a really strong combination play.
“He’s already had experience at his club in Uruguay and his goal-scoring record is quite impressive for a young man at 22. We hope we can give him the environment with Mason Holgate and other youngsters that we feel very, very strong about.”
Martinez is still keen to add another two attackers, as well as Funes Mori to supplement his defence, before the transfer window closes.