West Indies crashed to their fourth successive Super Eights loss which left them on two points and needing a miracle series of results to allow them to creep into the semi-finals.
"We are all disappointed and we are very sorry for the way we have performed in this World Cup," said Lara.
"All we can say is that we are sorry to our people."
The defeat will also increase calls for Lara, in his third spell at the helm of the team, to step down as skipper with a tour of England to commence just after the World Cup.
But he insists the priority now is for the team to win their last two matches - against Bangladesh and England in Barbados on April 19 and 21 - and restore a little bit of pride to the region.
"There will be no semi-final place for us but we have to play with pride and character in the next two games," said Lara.
"It's a chance to show we can perform with pride and character."
Lara won the toss on Tuesday and put South Africa into bat but then watched helplessly as AB de Villiers overcame injury and dehydration to hit 146 while Jacques Kallis (81), Herschelle Gibbs (61) and Mark Boucher (52) all took Graeme Smith's side to an imposing 356-4 in 50 overs.
The West Indies reached 279-9 in reply with only Ramnaresh Sarwan (92) getting beyond 50.
"We knew we had to win today, but things didn't work out," admitted Lara.
"But all credit to South Africa. They came back from a defeat against Bangladesh and played really well. Chasing 356 was always going to be tough.
"I can't fault the guys. They tried their best but we didn't come up with trumps.
"It's a huge blow not to reach the semi-finals. The guys are really down in the dressing room, it's a really quiet place. There are a lot of sad faces but we have to pick ourselves up and show that we can host a good World Cup."
The win kept South Africa, who now have six points, in fourth place in the Super Eights table with tough matches to come against unbeaten New Zealand and then England.
"It was a tough few days after playing so badly against Bangladesh in Guyana," said Smith.
"But we played our natural game under pressure. We got two points which was crucial. I also liked the way we played today, with passion and with freedom.
"We had the match won pretty early."