Sri Lanka had started the final day on 110 for one, optimistic of chasing down the 370 victory target after Amla had declared on 206-6, but collapsed under the pressure of the two quicks to lose nine wickets for 106 runs
South African fast bowlers Dale Steyn and Morne Morkel ripped through the Sri Lanka batting lineup to seal a 153-run win in the first test on Sunday as Hashim Amla started his captaincy in fine style.
Steyn took four for 45 and Morkel finished with four for 29 to dismiss the hosts for 216 shortly before the tea break and seal a first win in Sri Lanka for 14 years.
Steyn ended with match figures of nine for 99 to take the man of the match award after the best bowling figures by a quick at Galle.
"His figures speak for themselves, he bowled excellently," Amla said of Steyn.
"The lengths he bowled, the pace, the aggression - a great performance from a great bowler.
"I could look in the dictionary for a million adjectives to describe him, but I think it suffices to say that it was a great performance by a great bowler."
Steyn started South Africa's victory push by grabbing an early breakthrough on Sunday when he had Kaushal Silva edge to wicket-keeper Quinton De Kock to end his innings on 38 and the second wicket stand of 104 with Kumar Sangakkara.
Morkel then struck with the first ball after the drinks breaks when Mahela Jayawardene edged behind for 10 to leave Sri Lanka on 138-3.
The hosts still held hope, though, with Sangakkara holding firm against the barrage before the former skipper fell to one of the worst balls of the day.
The left-hander pulling a woeful long hop delivery from JP Duminy straight to Amla at mid-wicket to depart for 76 and leave the hosts in big trouble on 149-4.
Sangakkara could not believe what he had done, staring at the pitch before walking off with his head down.
Lahiru Thirimanne was the next to go, falling in similar circumstances to the first innings as he edged a ball from Steyn outside his stumps to AB de Villiers in the slips for 12.
Dinesh Chandimal then completed the miserable session by departing for one as he gloved an attempted pull off Morkel to de Kock to leave the hosts six down at lunch.
Steyn returned in the second session to take his fourth wicket when he had Dilruwan Perera caught by de Kock for his second duck of the match. It was the fourth wicket to go down for only 12 runs as the hosts fell to 161-7.
Duminy then ended Rangana Herath's 18-ball cameo for 20 when he top edged a sweep and was caught by a darting De Villiers, who ran from slip to short fine leg to take the catch.
Morkel wrapped up the victory by claiming the final two wickets in two balls, dismissing Suranga Lakmal at long leg for 12 and then Shaminda Eranga fending a short ball to short leg to leave Angelo Mathews unbeaten at the crease on 27.
"The bowlers were really hungry – Dale and Morne asked for the ball regularly, which was brilliant," said Amla.
"We picked up a few early wickets which made it a lot easier to maintain the pressure."
Sri Lanka captain Angelo Mathews blamed his middle order batting for the defeat.
"Where we lost the test was the failure of our batsmen to score runs on the first innings," said Mathews, who reckoned Sri Lanka were at least 100 runs short in their first innings when they folded for 292.
"When we started the day at 110 for one, our plan was to go after the target but like in the first innings we didn't bat well enough."
The second test starts in Colombo on July 24.