Leeds were cruising at 2-0 ahead going into injury time of Sunday's quarter-final when they were rocked by a West Ham double that forced extra time. However, they held their nerve to win a penalty shootout to book a last-four encounter with Premier League rivals Chelsea.
Southampton's reward for beating league leaders Arsenal is a semi-final tie with Manchester City, who beat Liverpool 4-0 in their quarter-final on Saturday.
The ties will be played at Wembley on the 25th and 26th of April.
"It's always going to be a difficult match against Chelsea," Dominic Calvert-Lewin told TNT Sports after the draw.
"They're a good side, but it's the FA Cup, we're going to Wembley. Excited. I've never had the opportunity to play at Wembley for club before, so for me it was a big chance today.
"When it went to 2-2, I thought maybe it's not meant to be, but thankfully, we stuck in there and got the win."