John Terry was one of eight changes to the Chelsea team that started against Manchester City on Sunday and the heavily-changed side made a slow start against an out-of-sorts Boro, who have now won just one of their last seven games in all competitions.
In an uninspiring first half Oscar came close to breaking the deadlock when his effort from close range was well blocked by Andre Bikey, while Scott McDonald headed wide and Mustapha Carayol drove a long range effort over at the opposite end.
Chelsea came back after half-time in determined fashion and took the lead six minutes after the restart when Ramires' curling shot deflected off Fernando Torres and nestled in the top corner.
Neat link-up play between Oscar and Eden Hazard allowed Moses to tap the ball into an empty net on 73 minutes as Rafa Benitez's side ended Boro's hopes of a cup upset.
The home side threatened to pull a goal back when McDonald struck wide from 20-yards and Merouane Zemmama tested Petr Cech with a free-kick, before Ishmael Miller squandered two opportunities to put Boro back in the game late on.
The win moves Chelsea into the FA Cup sixth round where they will meet Manchester United at Old Trafford on March 10.