The pair of Nottinghamshire bowlers took nine wickets between them as South Africa, who began the day on 76 for six, were all out for 133 after 18 overs on the final day.
Swann claimed five for 54 and Broad took four for 43 to help England take a 1-0 lead in the four-match series.
"It's not far off the best performance I've seen by an England team," said England captain Andrew Strauss, "but I'm aware the series is very much alive and there is still hard work to be done."
South African captain Graeme Smith admitted his side had been outplayed.
"We've got to look at ourselves in the mirror and bounce back very quickly," he said.
The third Test starts in Cape Town on Sunday.
Off-spinner Swann had match figures of nine for 164 and earned his second successive man of the match award following a five-wicket haul and an innings of 85 in the drawn first Test in Centurion.
South Africa's faint hopes of extending the match beyond lunch evaporated when Morne Morkel was out to the 11th ball of the morning.
Morkel was dropped by Strauss off the previous delivery from Swann, which the left-hander edged low to the England captain's right hand, with the ball going on for four.
But Swann pushed the next ball through quicker and Morkel, stuck in his crease, was trapped leg before wicket.
Mark Boucher went on to make 29 before he gloved Stuart Broad down the leg side to fellow wicketkeeper Matt Prior.
He was given not out by umpire Amish Saheba but England asked for a review and replays showed clearly the ball had deviated off his glove so Boucher had to depart for the second time in the match after initially being given not out.
Paul Harris hit 36 before he scooped James Anderson to Broad at mid-on and Swann completed a comprehensive win when he had Dale Steyn leg before wicket.
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