France international Giroud struck twice in as many minutes and there was no way back for Championship high-flyers Boro, who were outclassed at the Emirates Stadium.
Aitor Karanka's side sprung a major surprise by winning 2-0 at Premier League champions Manchester City in the fourth round, but never looked like repeating that feat as Arsenal taught them a lesson.
Striker Giroud rounded off a lovely Arsenal move to open the scoring and capitalised on some slack defending to double the London club's lead and keep them on course for another trip to Wembley in May.
Boro had won six consecutive games and not lost in 10 ahead of their trip to the capital, but must now turn their attention back to the race for promotion to the Premier League after they were given an indication of the task they will face if they return to the top flight.
Gabriel Paulista made his Arsenal debut and leading scorer Alexis Sanchez was passed fit as Arsene Wenger made seven changes to the side that beat Leicester City 2-1 on Tuesday.
There were eight alterations to the Boro side following their midweek win over Blackpool, on-loan striker Patrick Bamford - who scored the opening goal in the stunning victory at City - among the players restored to the side.
Arsenal had won their previous eight home games and were brimming with confidence as Boro struggled to contain them in the opening stages.
Santi Cazorla forced an early save from fellow Spaniard Tomas Mejias with a measured free-kick and the Boro goalkeeper dived to his left to palm away a first-time strike from Giroud that looked to be heading just wide of the far post 13 minutes in.
Mejias was unable to prevent the France striker from opening the scoring after 27 minutes, though, when Cazorla's incisive pass found Kieran Gibbs in space and his first-time cross from the left was turned in by Giroud to round off a slick move.
There was little Boro could have done to avoid that well-worked opening goal, but the Teesside outfit were caught napping when Giroud struck just two minutes later.
The former Montpellier frontman escaped his marker and showed great technique to volley home Sanchez's corner with a clinical left-foot finish from six yards out.
Sanchez almost got in on the act when Mesut Ozil picked him out and the Chile forward beat Mejias, but Adam Clayton was on hand to clear as the ball trickled towards the Boro net.
Arsenal carried on where they left off at the start of the second half and Ozil shot straight at the busy Mejias before Gibbs stung the palms of the former Real Madrid keeper with a strike from a tight angle as the home side continued to dominate.
Sanchez's glancing header looked destined for the far corner of the net until Mejias produced a fine save to thwart the former Barcelona man, who had rose to meet Cazorla's inviting free-kick after 64 minutes.
Mejias then denied Theo Walcott when the England forward raced clear five minutes from time and Kike came close to a consolation when his stoppage-time header came back off the post as Arsenal eased into the last eight.
