The home team, with Frank Lampard back in the dugout for a second spell, came into the match at Stamford Bridge with form and history against them.
They started brightly, playing enterprising football, but rued two huge missed chances in the first half that would have heaped pressure on the defending champions.
The Spanish giants made Chelsea pay for their profligacy when Rodrygo scored in the 58th minute and he doubled his tally with 10 minutes to go.
Chelsea, the 2021 European champions, have now lost all four of their matches since Lampard returned to the club as caretaker boss earlier this month in place of the sacked Graham Potter.
Real Madrid manager Carlo Ancelotti stuck with the same team that won the first leg 2-0 last week while Chelsea caretaker boss Lampard restored Reece James, Mateo Kovacic and N’Golo Kante to his line-up.
Roared on by an expectant crowd despite their current woes, the home side made a bright start against the 14-time winners.
They had a glorious chance to take the lead but Kante skewed his left-foot shot wide from about 12 yards.
The visitors, content to bide their time, looked sharp on the break.
Midway through the opening period Rodrygo created space for himself and thumped an effort against the outside of the post from an acute angle, though goalkeeper Kepa Arrizabalaga may have had it covered.
Luka Modric tested Kepa with a shot from an angle and five minutes before half-time Vinicius Junior skewed a shot wide from close range after a cross from the Croatia midfielder.
But Chelsea should have been ahead on the stroke of half-time when Reece James fizzed a ball across the box from the right and it fell to Marc Cucurella at the back post.
The defender took a touch and shot but Thibaut Courtois came flying out of his goal to save.
Kante had another golden opportunity to take the lead early in the second half but his close-range effort hit Eder Militao and deflected away to safety.
Real took the wind out of Chelsea’s sails with just over half an hour to go, when Rodrygo scored from close range after an assist by Vinicius.
Lampard made a number of attacking subsitutions, bringing on Raheem Sterling, Joao Felix and Mykhailo Mudryk in a desperate bid to get back into the match.
But Rodrygo found the net again in the 80th minute, tapping home after a clever assist from Federico Valverde.
Real Madrid will likely play Manchester City in the semi-finals, with Pep Guardiola’s team 3-0 up against Bayern Munich after the first leg of their last-eight tie.
But Chelsea are contemplating the final weeks of a disastrous campaign despite a staggering spend of more than £500 million ($621 million) in the first year of the club’s new ownership.
Already out of both domestic cups, they are languishing in 11th place in the Premier League and look unlikely to qualify for European football next season.
(With inputs from AFP)