Chasing just 134 to win, the Proteas reached their target in the 18th over with opener Quinton de Kock making an undefeated 78.
Sri Lanka, who won the toss, were dismissed for just 133, with leg-spinner Imran Tahir taking 4-26 and off-spinner JP Duminy 3-29, including a hat-trick.
“I’m quietly confident. I’m always a positive kind of guy and I always believe in big things,” said De Villiers, whose side play either New Zealand or the West Indies in the last four.
“I spoke to (fast bowler) Morne Morkel, who loves fishing, and I said it’s almost like you’ve got the big fish on the hook and you’ve got to reel it in.
“We had a really good feeling when we came to the ground today. We were almost obsessed with our goal.
“We always knew JP would probably have to bowl 10 overs, so for him and Imran Tahir to come on in a crucial stage of the game and perform like that gives the spinners a lot of confidence.”
Duminy became the first South African to take a World Cup hat-trick and picked up the man-of-the-match award.
“I got everything from South Africa. I always feel I have to pay the country back,” he said.
“I’m just really pleased and honoured to be part of this team, playing with some great players. I’m just trying to do my role really. It’s an absolute honour and I’m really loving it.
“I really don’t know what to say. I’m pleased that all my hard work paid off today, what I was thinking about this game and stuff. So yeah, really glad.”
Sri Lanka captain Angelo Mathews admitted his team suffered severe stage fright as they crashed out of the tournament.
He said: ???A score of 130 was not much. There were no demons on this track. But we didn’t bat well. Maybe it was nerves, we didn’t go for our shots and it’s easily our worst performance we have done in the World Cup and it happened to be in the quarter-final.???
Kumar Sangakkara and Mahela Jayawardene were both playing their final one-day international for Sri Lanka.
“It’s disappointing to not make it a memorable one for Mahela and Sanga,” said Mathews.
“On behalf of the nation and the team I thank them for their invaluable service. The best thing for us would have been to go to the final and win it for them but it was not to be.”