Once Kyle Abbott dismissed Sri Lanka captain Angelo Mathews for 59 in the third over of the day in Port Elizabeth, the tone was set.
Abbott (2-38), Kagiso Rabada (3-77) and Keshav Maharaj (3-86) all played their part as they went 1-0 up in the series.
Stephen Cook's second-innings 117 had set the home side up for the win.
South Africa are on track for a third-straight series win after losing their number one ranking at the start of the year.
The second game in the three-match series starts in Cape Town on 2 January.

The one-year-old was handed in to the Dogs Trust near Darlington in December after being found wandering the streets as a stray.
Following an appeal on social media, the canine with "a heart of gold" has found a new forever family.
The trust said Fester would be leaving the charity's kennels early next week.
Fester was born with a narrow jaw which causes his lower teeth to stick out - creating some "unique features", a trust spokeswoman said.
"We were concerned that fun-loving Fester may have ended up being over-looked by potential new owners due to his unusual eye-catching appearance," she added.
"He is a gorgeous boy with an amazing character, so we couldn't imagine why anyone wouldn't want to have him as part of their family.
"He's had a rough time and it is fantastic news he will have a new home where he can settle and hopefully his character will shine through."
South Africa beat Sri Lanka by 206 runs thanks to a five-wicket haul in the first 14 overs on the final day of the first Test.
[NEXT_CONCEPT]
Abandoned because of his bizarre looks, Fester the boxer dog, who is blind in one eye and has a protruding lower jaw, has finally been found a new home.