The Frenchman scored twice in three minutes from close range to give Arsenal a comfortable lead at half-time, and that’s how it stayed as Champiosnhip high-flyers Boro failed to make a mark on the match.
The result means Arsenal have reached the last eight for only the fourth time in 10 years, and Arsene Wenger’s side will find out who they face in the next round on Monday.
Wenger handed a debut to Brazilian defender Gabriel Paulista, and also gave goalkeeper Wojciech Szczesny a start in goal in place of David Ospina.
Middlesbrough, who had knocked out Manchester City at the Etihad Stadium in the last round, struggled with possession early on, clocking up just 32 per cent in the first 45 minutes.
Goalkeeper Tomas Mejias was busy throughout, doing well to keep out Santi Cazorla’s free-kick in the sixth minute.
As Arsenal probed, they eventually took the lead in the 27th minute through Giroud, slotting home with his left foot inside the six-yard box after Kieran Gibbs’ cross.
Giroud added a second two minutes later, darting into the near post area to glance home Cazorla’s corner with the inside of his boot.
Questions will be asked of Boro’s defending, but the 28-year-old finished superbly for his 10th goal of the season.
Mejias had to be alert to save from Giroud again before half-time, while Alexis Sanchez failed to get the better of the Spanish stopper when put through on goal by Mesut Ozil.
The hosts took their foot off the gas in the second period, but forced a fine save out of Mejias again, saving low to his left from Sanchez’s glancing header.
Substitute Theo Walcott should have added a third with two minutes remaining after being put through on goal by Tomas Rosicky, but he dragged his shot for Mejias to parry wide.
Middlesbrough left it late to rally, seeing Albert Adomah flash an effort from 20 yards just wide, before Spanish forward Kike hit the post with a close-range header.