Garcia watched his side crumble to a 4-1 loss in the second leg at Pride Park on Sunday, and a 6-2 aggregate defeat in the tie overall.
The Spaniard, who took over from Gus Poyet following his resignation last May, cut a dejected figure after his team let promotion hopes slip through their fingers with a whimper.
And he has now told the south coast club that he wishes to leave.
However, it is believed the club's hierarchy have asked him to take more time to think things over ahead of further talks over his future in the coming days.
Brighton, who lost in the play-off semi-finals under current Sunderland boss Poyet last season, crept into this season's competition with a splurge of good results at the end of the campaign.
Garcia's side won 19 of their 46 games overall, and they fought gallantly into a 1-0 lead against Derby in the first leg at the Amex Stadium on Thursday, before seeing that overturned by Steve McClaren's in-form side.
And they could manage only a late consolation in the return fixture on Sunday as Will Hughes, Chris Martin, George Thorne and Jeff Hendrick fired the Rams to a Wembley date with either QPR or Wigan on May 24.