Giroud ensured Chelsea led with the help of slack goalkeeping from De Gea at the end of a half in which United lost Eric Bailly to a serious-looking heading injury.
De Gea was at fault again almost immediately after the restart when Mount doubled Chelsea's lead with a shot a keeper of any level would hope to save.
The Spaniard was not entirely convincing when Harry Maguire put into his own net to compound a miserable day either, but – despite a late penalty from Bruno Fernandes – Frank Lampard's side were good value for the win and can now look forward to a return to Wembley to face Arsenal in a rematch of the 2016-2017 final.
Chelsea's pressure had United on the back foot early on but Reece James' stinging drive was the only save De Gea had to make, with Marcos Alonso wastefully putting a free header over.
United's performance was at odds with its recent form and it saw Bailly taken away on a stretcher in a neck brace after a nasty clash of heads with Maguire.
After a lengthy delay, Giroud's deft touch from Cesar Azpilicueta's right-wing cross squirmed past De Gea, who might feel he should have done better, in the 11th minute of first-half stoppage time.
Just a minute after the restart De Gea's outing got even worse when he allowed Mount's seemingly unthreatening 20-yard effort under his body, albeit Brandon Williams's wayward pass did his side no favours.
Marcus Rashford flashed a tantalising ball across the face of goal and Maguire headed wide when Willy Caballero flapped at a corner as United tried to muster a response.
But it was game over when Maguire got the final touch on Alonso's drilled cross and De Gea was again beaten, with Fernandes' clinical spot-kick after Callum Hudson-Odoi's clumsy challenge on Anthony Martial providing scant consolation.