Zimbabwean Khama Billiat put Ajax ahead on the half-hour mark, before Abafana Bes'thende substitute Siphesihle Nzimande equalised on 60 minutes.
The visitors began to take control just past the 15-minute mark when home keeper Helton da Rocha was forced to parry Billiat's thunderous shot from close-range, a magnificent save, but a sign that Ajax were beginning to take control of the game.
And the Urban Warriors did take the lead on 29 minutes when Billiat netted after being put through by Thulano Serero, keeping his composure with just Da Rocha to beat.
Lebogang Manyama then provided Billiat with another opportunity seven minutes before the break, but Da Rocha was smartly off his line to smother the danger.
Golden Arrows went in search of the equaliser after the break and they were rewarded when after Thanduyise Khuboni had won a free-kick, Mkhize delivered the set-piece and halftime substitute Nzimande was on hand to head home from close-range.
It was all the home team after that as Siyabonga Nkosi headed just wide from an Nzimande cross and substitute Murakami had his goal-bound shot blocked by an Ajax defender from close-range.