Victor Moses scored the winner for Chelsea
Chelsea will face Manchester United in the FA Cup quarter-finals after comfortably beating Championship side Middlesbrough at the Riverside, courtesy of goals by Ramirez and Victor Moses.
Chances were limited on a freezing night on Teesside as Scott McDonald headed wide for the hosts in the first half before Petr Cech spilled Mustapha Carayol's shot, reports the BBC.
Chelsea made the breakthrough after the interval when Ramires's shot hit Fernando Torres on its way in.
Eden Hazard then came on to add more thrust to the visitors' attacking play, and the Belgian's clever one-two with Oscar helped set up Moses for a simple finish after 72 minutes.
Boro did belatedly respond with substitute Merouane Zemmama testing Cech with a late free-kick, Nicky Bailey heading wide, and Ishmael Miller volleying over from a good position.
But prior to conceding, Tony Mowbray's team failed to regularly test a much-changed Chelsea side, in keeping with a wretched run of one win in nine league games this year.
Torres had a couple of surging runs in the first period without reward, but he was in the right place on 51 minutes as Ramires's 25-yard shot flicked in via the Spaniard on the edge of the area, following neat work by Oscar.
The goal was tough luck on Middlesbrough, but once Hazard was introduced Chelsea attacked with more purpose, the Belgian and Oscar both shooting wide as the visitors pressed.
Hazard was involved in the Blues' second goal - a delightful move that culminated in a straightforward finish for Moses, who earlier in the month won the African Cup of Nations with Nigeria in South Africa.
Victory for Chelsea, currently fourth in the Premier League, keeps them in the hunt for two trophies - they are challenging for the Europa League, as well as the FA Cup - in addition to their quest for Champions League qualification.