South Africa were all out for 83 after Broad (right) tore through their order
Broad blew the Proteas away with a stunning spell of 5-1 in 31 balls as England took an iron grip on this third Test.
He then finished things off with a brilliant diving catch off his own bowling as the hosts were routed for 83 in their second innings.
England, who had taken a 10-run first-innings lead after making 323 earlier in the day, then chased down 84 runs inside 23 overs to set the seal on a crushing seven-wicket victory.
It saw Alastair Cook’s men take an unassailable 2-0 lead ahead of next week’s final Test in Pretoria, securing England’s first overseas series win since beating India in 2012.
Cook said: “It’s pretty special feeling and I’m looking forward to getting back in the changing room to spend some time with a group of guys who’ve done something very special.
“You can look at moments in the series where we fought really well, were tested and came through it. To win two Test matches out of three and dominate quite a lot of the other one is huge credit to the guys.”