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The left-handed Ingram made 105 not out as South Africa piled up 315 for four after being sent in to bat. Pakistan could only make 190 in reply.
South African captain AB de Villiers described the win as "a huge step in the right direction" after what he admitted had been "worrying and frustrating" recent one-day performances by the team.
Pakistan captain Misbah-ul-Haq said his side had been unable to exploit helpful bowling conditions early on.
Misbah said an opening partnership of 72 by Graeme Smith and Hashim Amla had set up the South African innings.
"The way the ball was seaming, they played really well. We were looking to get early wickets. If South Africa had scored 270 or 280 it could have been a good game," he said.
Ingram made his runs off 103 balls with ten fours. It was his third one-day international century and his second against Pakistan.
He shared partnerships of 120 for the third wicket with captain AB de Villiers (65), 62 for the fourth wicket with Faf du Plessis (26) and 50 off only 23 balls with Farhaan Behardien, who hammered 34 not out off 14 deliveries.
Smith and Amla set an aggressive tone with their opening partnership stand, with both batsmen going for their shots against Junaid Khan and Umar Gul despite the Pakistan new ball bowlers getting some early assistance from the pitch.
The South Africans opted for the batting power play immediately after the compulsory first ten overs but lost Smith when he gloved an attempted reverse sweep to be caught behind off Saeed Ajmal, one of four spin bowlers in the six-man bowling attack, for 30.
Amla made 43 before he was caught at deep square leg off Mohammad Hafeez.
De Villiers and Ingram then set South Africa on the way to a big total with a run-a-ball stand during which they kept the score moving quickly against the Pakistan spin bowlers with deft placement and good running between wickets.
Pakistan made a bright start with Mohammad Hafeez and Nasir Jamshed putting on 42 for the first wicket off 43 balls before they suffered a double blow.
Jamshed was caught at slip by Graeme Smith off Rory Kleinveldt for 25 and Hafeez fell for the same score in unlucky fashion when a drive by Younis Khan was deflected by bowler Kleinveldt into the stumps for Hafeez to be run out.
Asad Shafiq fell cheaply and Pakistan were unable to build substantial partnerships with captain Misbah-ul-Haq top-scoring with 38.
Shahid Afridi hit 34 off 16 balls before he was last man out but his innings was never likely to alter the outcome. Afridi hit three sixes, including a massive strike which sailed out of the ground and into a car park.
Kleinveldt, whose first two balls were hit for four, came back strongly to finish with four for 22 in 5.2 overs, while Ryan McLaren took three for 19.
The win gave South Africa a 1-0 lead in the five-match series. The second match will be in Centurion on Friday.
De Villiers said South Africa would be looking to build on Sunday's result. "We want to get consistency and we?ve got the momentum now."
Misbah said Pakistan need to work on their "death" bowling and fielding. "We need to cut down 30 or 40 runs (by the opposition)," he said.