Arsene Wenger's side were ahead inside seven minutes when Mesut Ozil shrugged off recent criticism to slot a low ball past Everton goalkeeper Joel Robles.
It was a much-needed confidence boost for the £42.5million midfielder, who was jeered by some German supporters for his performance in Wednesday's friendly against Chile, and who had not scored since the 1-1 Barclays Premier League draw against Everton in early December.
But the visitors rallied and went into the break level following a fine counter-attack down the right by Ross Barkley, which ended up with Romelu Lukaku bundling home from close range on 32 minutes.
Arsenal regrouped following the restart and were awarded a penalty on 67 minutes when Ozil's backheel released Alex Oxlade-Chamberlain down the right and he was tripped by Gareth Barry after cutting inside.
Former Everton and Rangers midfielder Mikel Arteta smashed home the spot-kick into the bottom left corner - only for referee Mark Clattenburg to order a retake for encroachment by Giroud.
Arteta kept his nerve to fire the ball past Joel again, this time high into the right side of the net.
With the tie having opened up, both sides made changes as Aiden McGeady and Leon Osman came on for Everton as Oxlade-Chamberlain made way for Tomas Rosicky, who has just signed a contract extension with Arsenal.
The Gunners secured their semi-final trip to Wembley when Bacary Sagna got away down the right and to the goal-line, where he cut the ball back for Giroud to crash home a third on 83 minutes.
And Arsenal saved the best for last as a blistering counter-attack ripped Everton apart. Santi Cazorla fed Rosicky down the right, who in turn picked out Ozil, and his first-time ball into the centre was swept home by Giroud to keep Arsenal on course for a first trophy since 2005.