The spiky, gripping derby clash turned against Diego Simeone's side in the additional period when defender Stefan Savic was sent off.
The build-up to the derby was tainted when a crude effigy of Vinicius was hung from a bridge near Real Madrid's training ground, which the club described as "a disgusting act of racism, xenophobia and hatred."
Madrid fans had planned to sing Vinicius' name in support in the 20th minute, matching his shirt number, but were stunned into silence by Morata's goal moments before.
Atletico, hitting their best form of the season, conjured a brilliant move to break the deadlock, with Morata tapping home after Koke's dink allowed Nahuel Molina to stroke a first-time cross to the back post.
Eder Militao might have equalised but headed over from point-blank range, the ball deflecting slightly behind his leap to make the chance harder.
Antoine Griezmann was moving with freedom between the lines, relishing his new deeper role and pulling the strings for Atletico as they dominated the first half.
Madrid added another name to a long injury list when Ferland Mendy had to be replaced before half-time, with Eduardo Camavinga slotting in at left back and Dani Ceballos coming on.
Along with injuries for Dani Carvajal, David Alaba and Lucas Vazquez, it left Madrid without any natural full backs.
Benzema was not far away from reaching Fede Valverde's flick-on from Nacho's drive, as Madrid started the second half strongly.
Jan Oblak produced a fine save to deny the French forward, with Vinicius unable to react quickly enough to finish from the rebound.
At the other end, Axel Witsel sent an overhead kick narrowly wide and Thibaut Courtois tipped a vicious Griezmann free kick over as Atletico tried and failed to kill off the game.
Rodrygo levelled with 11 minutes to go with his jinking solo effort and subtle near-post finish, taking the game to extra-time.
Atletico defender Savic foolishly handed Madrid the edge when he was sent off for two yellow cards in two minutes.
The first came during an argument with Vinicius and the second for fouling Camavinga.
Madrid immediately began to capitalise when Vinicius crossed for Antonio Rudiger, who headed narrowly wide.
Benzema struck lethally after 104 minutes when Vinicius miskicked Marco Asensio's cross and the ball travelled through to the striker.
Madrid, who last won the Copa del Rey in 2014, kept Atletico at bay in the final 15 minutes before Vinicius dribbled through and fired home to clinch the victory.
Earlier Athletic Bilbao won 3-1 at Valencia to join record 31-time winners Barcelona and Osasuna in Monday's semi-final draw.