Josephs was shown his marching orders after losing his head in the 78th minute during a scuffle that resulted from an off-the-pitch incident between substitute Judas Moseamedi and Gabadinho Mhango. But by that time Wits were already 3-0 up so no harm was done.
After a bright start by Wits, who did all the attacking, Eleazar Rodgers gave the Clever Boys the lead in the 22nd minute with a neat finish at the far post after being deftly picked out by the ever-lively Mhango, who had been causing trouble for the City defenders since kick-off.
Eric Tinkler's side struggled to create much in the final third and were then made to pay for some sloppy work in defence. Goalkeeper Shu-Aib Walters inexplicably came out of his box to take a throw-in and gave a dangerous pass to centre-back Tshepo Gumede, who was under pressure and had to put the ball back into touch.
That needless giveaway was pounced on by Wits, who took the throw quickly before Daine Klate found Elias Pelembe free in the area. The Mozambican waltzed past two flat-footed defenders and buried the shot past Walters, who will have wanted the ground to swallow him up.
Mhango made it 3-0 just before half-time. Having threatened on several previous occasions, the Malawian finally got the goal he deserved after latching onto Gumede's horrendous back-header and lobbing Walters, who was in no man's land.
Wits were full value for their lead having been by far the better side in the first 45 minutes, while City have a lot of thinking to do about the manner in which they conceded their goals.
The second half turned into a rather scrappy affair as neither side managed to hold onto the ball for long periods, and very few further chances were created.
Sifiso Myeni came on to make his second Wits debut in the 72nd minute and then six minutes later there was a moment of madness from Josephs that got the goalkeeper sent off.
Mhango was on the receiving end of a strong challenge from Moseamedi after the ball had gone out of play, which resulted in a scuffle. Josephs ran all the way out of his goal to defend his teammate and struck Roland Putsche in the face as they squared up.
Darren Keet replaced Mhango in the 80th minute for his Wits bow as Gavin Hunt's side comfortably saw out the remainder of the game.
Bidvest Wits: Josephs (sent off 78'), Allie, Hlanti, Hlatshwayo, Mkhwanazi, Motshwari, Pelembe (Myeni 72'), Monare (Gome 66'), Rodgers, Klate, Mhango (Keet 80')
Cape Town City: Walters, Kobola, Gumede, Rode, Seedat, Matsi (Putsche 46'), Sim (Adonis 76'), Manyama, Jayiya (Moseamedi 46'), Ngoma, Majoro