Arrows had led with five minutes to go, but a straight red card shown to Maximillan Mbeva - who had been the standout player in the match - changed the outcome.
Wits had come into the game unbeaten in seven, and it looked a mere formailty that they would stretch that lead after a strong start.
Mbeava stood strong to save Elias Pelembe’s shot after just four minutes, and the Namibian international also kept Sibusiso Vilakazi and Pkahamani Mahlambi at bay with some excellent keeping.
Arrows had a lot of the ball, but their only real chance came through the brilliant Kuda Mahachi, whose fierce shot that hit the side-nettting after a great solo run.
Vilakazi had the ball in the back of the net in the 37th minute, but he was rightly ruled offside.
Mbeava again denied Mahlambi just after the restart, before Arrows took the lead through Deon Hotto – the Namibian heading into the bottom corner after a fine Mahachi cross.
In the 73rd minute Wits found their equaliser through Mahlambi, with the Under-23 international’s first-time volley from Klate’s cross giving the keeper no chance.
But just three minutes later and Arrows were back in front when Mabhuti Khenyeza side-footed Andile Fikizolo’s square-ball into the net from the edge of the area.
Then the moment that changed the game: Mbeava was left exposed by his defence and was forced to save Mah;ambi’s goalbound shot outside his box.
He was sent off and replaced by Ricardo Goss, who picked the ball out the back of his net twice inside a minutes, with Klate getting both goals.
Klate scored from the resulting free-kick after Mbeava’s sending off, and a minute later had the winner with a sweeping finish into the far corner to ensure Wits extend their lead at the top of the standings.