Career
River Plate
Born in Montevideo, Rodríguez began his career with River Plate Montevideo. On 30 September he scored his first goal, opening a 2–0 home victory over Liverpool. He ended his first season with three goals in 24 appearances.
On 11 May, he was sent off in a 0–2 home defeat to Racing, along with teammate Sebastián Taborda.
In the second leg on 25 September, he equalised for a 1–1 draw, but his team were eliminated 3–2 on aggregate. The 2014-2015 season was Rodríguez"s most successful for River Plate, yielding nine goals in 23 games, including his first two-goal haul, in a 4–0 home victory over Rampla Juniors on 8 March 2015.
Everton
On 28 August 2015, Rodríguez signed a four-year contract with Everton of the Premier League, for a reported fee of £500,000. Manager Roberto Martínez said that he would likely be put into the first team, due to his experience of top-flight football in his homeland.
On 17 March 2016, Rodríguez joined Championship club Brentford on a one-month loan.
He made his debut three days later in a 0–1 home loss to Blackburn Rovers, playing 76 minutes before being replaced by Scott Hogan.