South Africa beat defending champions England 15-6 in the final in Paris on Saturday, a match watched at midnight here by the Proteas who are on a tour of Pakistan.
"I'm glad we were able to watch the match," said Smith.
"We're very proud of the rugby team. We rejoice in their victory along with everyone else in the country. They deserve everything that comes their way."
Smith, whose team won the two-Test series 1-0 for their first win in the subcontinent for seven years, was speaking after losing the second one-day international against Pakistan by 25 runs here on Saturday.
The Springboks' win was their second World Cup triumph. However, their cricketing counterparts have still to reach a World Cup final in five attempts.
South Africa coach Mickey Arthur said the Springboks win was a reward for hard work and determination.
"It shows what consistency of selection and planning for four years can do. I am pleased and proud," said Arthur.
Veteran paceman Shaun Pollock, a former skipper, said the win in Paris was 'fantastic'.
"It's exciting to be a supporter and to experience the winning of a World Cup. The heroes' welcome that they get is nothing less than they deserve," said Pollock.
Opening bowler Makhaya Ntini also hailed the success.
"Everybody played the sort of rugby that makes us proud," said Ntini.
Batsman AB de Villiers said he was anxious to join the celebrations back home.
"I was at school with Wynand Olivier and a house-mate of Bryan Habana. I can't wait to get home and have a beer with them," said De Villiers.