Tahir and paceman Chris Morris (2/32) were the best performers. Amla and fellow opening partner Quinton de Kock (23) struggled for runs initially, with Sri Lankan pacers Lasith Malinga and Suranga Lakmal keeping it tight from both ends.
Kagiso Rabada had spilled the simplest of caught and bowled chances off Upul Tharanga while fellow opener Niroshan Dickwella was smashing Wayne Parnell to all corners.
Then, in the 17th over, off the second ball, Pradeep set up newly arrived du Plessis with a short delivery.
Sri Lanka captain Tharanga has been suspended from the next two Champions Trophy matches after his side were found guilty of a serious over-rate offence.
Du Plessis made 75 in a second-wicket stand of 145 with Amla, who worked the ball off his pads and guided it behind square on the off side with the minimum of fuss.
Du Plessis went after the bowlers, allowing Amla to calmly reach his 25th ODI century, tied for fifth best all-time. "They have a lot of match-winners on their team", said De Villiers ahead of the clash. "He's got four or five days before the next game to get ready".
With these two quick wickets, South Africa were five down for 232 in 42.3 overs and still needed a strong finish.
"It also presented them with a selection dilemma", said the Sri Lankan great, "They choose the more conservative option and chose to bolster the batting, playing just four specialist bowlers with Asela Gunaratne as the fifth bowler".
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Tahir squeezed the Sri Lanka batting, and when captain Upul Tharanga, on 57, hit Tahir straight to a catch at deep cover at 146-5 in the 26th over, Sri Lanka's fate was certain.
Miller departed after the next ball, and a exhausted Amla soon followed suit, his despairing dive while coming back for a second was not enough to prevent Dickwella whipping off the bails after good work in the field by Mendis.
South Africa's fielding was outstanding as it produced two run outs through direct hits while the catching was top class.
Tahir then claimed Tharanga and Asela Gunaratne (4) to put his side in command.
The 25-year-old has played only six ODIs and the Sri Lankans believe his variety will trouble their opponents.
"They bowled exceptionally well, and credit to our openers for seeing it out".
South Africa scored just 44 runs in 12 overs when Quinton de Kock was caught behind off Nuwan Pradeep, and the medium-pacer nearly had his second wicket shortly after as du Plessis, on 8, skied a hook shot into the leg side.
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