The forward has emerged as a key part of Carlo Ancelotti's side this season but has been dealt a devastating injury blow
The Spain Under-21 international was withdrawn just three minutes into the second leg and Madrid have confirmed the initial fears that he will be out for a lengthy period.
A club statement read: "After scans and imaging tests conducted at the Hospital Sanitas La Moraleja, our player Jese Rodriguez has been diagnosed with a complete tear of the anterior cruciate ligament in his right knee."
Madrid went on to win the game 3-1 and complete a 9-2 aggregate triumph, yet the achievement of reaching the quarter-finals was overshadowed.
Ancelotti told AS: "It's a big loss. We send him all our best wishes."
Cristiano Ronaldo, who scored twice to draw level with Ferenc Puskas on 242 goals for the club - good enough for fourth on the all-time list, said: "I send much encouragement for a very important player for us.
"Jesé did very well when he played and his injury is bad for the team. It is very bad news for us."
Real Madrid's director of institutional relations Emilio Butragueno said: "It looks like the news isn't good [about Jesé].
"It's left everyone pretty upset."
Defender Sergio Ramos added on the UEFA.com Twitter feed: "The injury spoils this victory a little."
The injury will be a crushing disappointment for the player who has established himself as a reliable squad member this season.
The 21-year-old has scored five goals in 18 La Liga games to help Madrid open up a three-point lead over city rivals Atletico at the top of the table.