The Blues have set up a glamour date with rivals Manchester United in the next round whom they will meet at Old Trafford on Sunday, March 10.
The deadlock was broken just six minutes into the second half when Yossi Benayoun slid the ball to the top of the box with Ramires firing home past Boro keeper Jason Steele thanks to a deflection off Fernando Torres, which he nothing about.
With just over 15 minutes remaining, Moses gave the visitors an eventual unassailable lead when he poked home from close range from Oscar's pass to send the Southwest Londoners through to the final eight.
The win will ease some of the pressure building on interim boss Benitez who chose to start skipper John Terry just a day after playing down talk of a rift between the pair.
Chelsea are now sitting fourth on the Premier League table but still have an opportunity in both the Europa League and FA Cup whilst Tony Mowbray's Boro will now focus on forcing their way into a promotion playoff position as they sit seventh on the tier two table. 2012/13 FA Cup fifth round
Riverside Stadium, Middlesbrough - 27,856