A seesaw struggle between the top-ranked rugby nations took a decisive turn on 78 minutes when the Springboks were awarded a penalty try to build a 12-point advantage.
The rest of the South African points came from the boot of fly-half Handre Pollard with two conversions and two penalties. In an uncharacteristic display, he also missed three shots at goal.
Impressive debutant full-back Jamie Osborne and replacements Conor Murray and Ryan Baird scored for Ireland, with fly-half Jack Crowley slotting a conversion and a penalty.
Both teams finished with 14 men after late yellow cards for Arendse and replacement Ireland hooker Ronan Kelleher.
Victory completed a 'grand slam' for South Africa coach Rassie Erasmus who, before this Test, had plotted victories over every top 10-ranked country except Ireland.
South Africa began with 12 starters from the 2023 Rugby World Cup final triumph over New Zealand with injuries ruling out full-back Damian Willemse and loose-head prop Steven Kitshoff.
The other absentee was number eight Duane Vermeulen, who retired after the global showcase aged 37 having made 76 Test appearances for the Springboks.
Ireland last played in March, defeating Scotland narrowly in Dublin to become Six Nations champions, and they made two changes from that starting line-up.
Coach Andy Farrell chose Osborne at full-back over several rivals, including Jordan Larmour, a late inclusion against the Scots after Hugo Keenan was injured warming up.
The other alteration was enforced with a hamstring injury ruling star scrum-half Jamison Gibson-Park out of the tour. Craig Casey replaced him.
Ahead of the Pretoria showdown, South Africa had an 18-9 advantage over Ireland in victories with one match drawn in a 118-year rivalry.
The Springboks dominated the Irish for decades with 14 wins in 16 internationals, but the Irish turned the tables from late 2004, winning eight of 12 Tests before this tour.
Dramatic start
Starts to Tests do not get much better than the one South Africa made before a sell-out 52,000 crowd as Arendse scored just 149 seconds into the opening half.
The ball moved swiftly from one side of the pitch to the other and captain Siya Kolisi sent Arendse clear to dot down. Pollard converted.
Ireland responded immediately with pressure that took them inside the 22, but a knock-on by lock Joe McCarthy brought relief to the home side.
Crowley and Pollard exchanged penalties to leave the Springboks 10-3 ahead midway through a half notable for the desire of the world champions to run with the ball.
Another Pollard penalty stretched the lead to 10 points before Crowley surprisingly failed from his second shot at the posts.
Irish disappointment was soon forgotten, though, as Osborne scored in the corner after a bridge pass from hooker Dan Sheehan and a touch of brilliance by winger James Lowe.
As New Zealand-born Lowe was about to be bundled into touch, he calmly passed to Osborne, who dived over. Crowley missed the conversion, leaving South Africa 13-8 ahead at half-time.
After Crowley fluffed two penalty attempts, the same fate befell Pollard, just before the break and early in the second half.
A Lowe try was disallowed because Kelleher handled the ball in a ruck, Pollard missed a third consecutive penalty and Casey was stretchered off as drama unfolded.
Kolbe struck on 65 minutes, kicking the ball forward to score after Lowe had acrobatically prevented the ball going out of play. Pollard converted and South Africa were 20-8 ahead.
In a frenetic finish, scrum-half Murray and loose forward Baird scored, either side of the South African penalty try. Crowley converted the first.
The second and final Test will be played in Durban next Saturday.
