The debutant took 3-2 in four overs as England were reduced from 205-4 to 228-9, but Smith's side could not grab the final wicket that would have given them victory in the first of four Tests.
"There's a touch of frustration maybe at not getting over the line, but definitely not disappointment," said the skipper.
"It was a fantastic way to finish a Test match and I'm proud of the way the guys played throughout the game. We've really given it everything."
Smith felt his side had probably let the chance of victory slip away on day three, when Graeme Swann shared in a ninth-wicket stand of 106 with James Anderson to get England to within 62 runs of South Africa's first-innings total.
"We'd worked so hard to get there and we just let it slip," said Smith. "But we hadn't played a Test match in nine months and we played very good cricket. We can only hold our heads up high."
He added: "I was proud of the way the guys stuck at it. After the new ball this morning, there really wasn't much available for us.
It shouldn't have been that close - Strauss
"We tried pretty much everything in the middle period and it took a great spell of bowling with the new ball from Friedel to give us a chance.
"I said to the boys at tea, it is a crazy game and it's important that we stick at it. We've really given everything the whole day and there's still three Test matches to follow."
Smith admitted he had some difficult choices to make for the second Test in Durban, which starts on Boxing Day, with de Wet proving so impressive and premier fast bowler Dale Steyn on the road to recovery after injury.
"Friedel handled himself well and bowled with great energy, control and intensity," said Smith. "It was impressive to see and it gives some exciting choices going into the second Test."
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