Steyn's figures, the best by a quick at the Galle ground, helped the visitors bowl out Sri Lanka twice in the match despite a wicket which offered little help.
It was only South Africa's third win on the island and their first since 2000.
Steyn was well-supported by seamer Morne Morkel, who picked up four wickets as Sri Lanka's second innings ended on 216 shortly into the second session.
Sri Lanka needed to make another 260 runs on the final day with nine wickets in hand to pull off a come-from-behind win. But they lost their remaining wickets for just 106 to go down tamely in the end.
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Amla celebrated his winning start by hugging his teammates and exchanging high fives.
South Africa can now reclaim their top world ranking from Australia if they win the second and final Test in Colombo from July 24-28.
Part-time spinner Jean-Paul Duminy picked up two wickets including the crucial scalp of Kumar Sangakkara, who made a fighting 76 to salvage some pride for the hosts.
Sri Lanka, who had resumed on their overnight 110-1, added just 51 off 24 overs in the first session while losing five wickets.
His victim was opener Kaushal Silva (38) who nicked the seamer behind the wickets, where Quinton de Kock dived to his right to take a stunning catch.
Silva's dismissal ended a 104-run stand with Sangakkara for the second wicket, the highest of the innings.
Sri Lanka were dealt a bigger blow off the first ball after the drinks break when Morkel had big-hitter Mahela Jayawardene caught behind for 10.