Jayawardene said the fact that his team came so close to winning after a poor batting performance showed the ability and the fighting spirit of his players.
"We dont give up. From any situation we believe we can win matches," he said.
Looking ahead to Sunday's match against hosts West Indies at the same venue, Jayawardene said: "We are a very professional unit and we know exactly where we went wrong. The guys will make sure that we come back very strong on Sunday."
For the Sri Lankans it will be a second opportunity to take advantage of conditions which Jayawardene admitted were akin to those found in their native country, with a slow pitch providing plenty of help for the spin bowlers, spearheaded by Muttiah Muralitharan.
It was Muralitharan who kept Sri Lanka in with a slight chance Wednesday, bowling superbly to take three for 34 before unorthodox fast bowler Lasith Malinga gave them an unexpected sight of victory by becoming the first bowler to take four wickets in four balls in a one-day international.
From needing only four runs with five wickets in hand, South Africa found themselves with their last pair at the crease and three still needed. A four by Robin Peterson settled South Africa's nerves.
Jayawardene said Malinga, who finished with four for 54, could be a big force in the World Cup.
"He usually picks up a few early wickets and he can be a handful for the lower middle order and the tailenders. I can use him at any stage. He's something special that we have in this team.
Jayawardene says there is no special action needed to spruce up the batting which failed to click on Wednesday after he had won the toss and elected to bat in the first international match to be played at the new stadium at Providence on the outskirts of the capital Georgetown.
"We batted really well in the first three games we played here (in the World Cup)," said the captain whose side made it through from the first round with a perfect three wins out of three.
South Africa captain Graeme Smith endorsed Jayawardene's view of Sri Lanka's prospects.
"Sri Lanka in these conditions are going to be a tough team to beat," he said.