Second-half goals from Ramires and Victor Moses secured victory at the Riverside Stadium as Chelsea bounced back from Sunday's 2-0 loss at Manchester City, which had seen them slip to fourth place in the Premier League.
Victory in the fifth-round tie means Chelsea remain on course for what would be a fifth FA Cup title in seven seasons.
Interim manager Rafael Benitez has complained about fixture congestion in recent weeks, but with his side also still in contention in the Europa League, the European champions look set for a busy few weeks.
Chelsea made eight changes to the side beaten by City at the weekend, with Fernando Torres and skipper John Terry both returning to the starting line-up.
Oscar and Moses also came back into the team and they almost combined to break the deadlock in the 11th minute, but Oscar could not get enough purchase on his shot from his team-mate's cut-back.
Chelsea started authoritatively but as the first half wore on, Middlesbrough came back into the game.
In the 26th minute, Scott McDonald came close to opening the scoring but his header from Nicky Bailey's right-wing cross drifted narrowly wide of the right-hand post.
The hosts went even closer five minutes before half-time, with Petr Cech spilling a shot from Boro winger Mustapha Carayol and obliging Terry to make a hurried clearance.
It therefore came as something as a relief to the visitors when Ramires put them in front six minutes into the second half.
Yossi Benayoun laid the ball back to Ramires and his placed shot from the edge of the box beat goalkeeper Jason Steele with the aid of a slight deflection off the back of the crouching Torres.
The goal calmed Chelsea's nerves and after Steele had tipped over a shot from Torres, Terry headed wide at a corner.
Eden Hazard replaced Benayoun and the Belgian quickly created Chelsea's second goal in the 73rd minute, neatly exchanging passes with Oscar and speeding into the box before squaring for Moses to score from close range.
The hosts produced a flurry of late chances, with Cech notably saving from Ishamel Miller, but Chelsea survived to take their place in the last eight and take some of the heat off the under-fire Benitez.