Two second-half goals from Ramirez and Victor Moses were enough to set up the quarter-final clash with United at Old Trafford.
Chelsea were held in the first half but after just six minutes into the second half Brazilian Ramirez lashed onto the ball from 20 metres and his shot took a huge deflection.
Substitute Eden Hazard was instrumental in the second goal. The Belgian raced through the left flank beating a tired Middlesbrough defence to set up Moses to double the lead.
Meanwhile, frustrated by fans criticism, Chelsea manager Rafa Benitez announced that he would leave the club at the end of the season. The Spaniard blasted the clubs hierarchy and supporters during the post-match press conference,
"I will leave at the end of the season - they don't need to waste time with me," said Benitez.
Benitez asid the title of interim manager given to him when he replaced sacked Roberto Di Matteo was a mistake.
"It's because someone made a mistake. They put my title interim manager and I will leave at the end of the season so they do not need to waste time with me, they have to concentrate on supporting the team," he said.
Fans were shocked when Di Matteo was sacked after leading the club to their first ever Champions League title but were furious when the club decided to appoint Benitez as their manager for the rest of the season.