The White Knights looked to be cruising when they grabbed an early lead before Pyramids' defender Ali Gabr was given a straight red card for a las-man foul after 11 minutes.
However, Moroccan midfielder Walid El Karti brought Pyramids level in the second half and the big-spending club prevailed in the shootout with a 4-3 win
Zizo was one of two Zamalek players to miss from the spot. In embarrassing fashion, his "Panenka" effort was saved by the outstretched leg of keeper Ahmed El-Shennawi.
"Just like when you are happy with the support and love, you must also accept the other bad side of life and you must be responsible" Zizo wrote on his official Instagram account.
"One day you can be the best person in the world and another people will forget everything you have done. That is why in such difficult times I have to apologise because I would never run away or hide. You are free to accept or reject it but I have to do it."
The 27-year-old has been Zamalek's outstanding performer over the past couple of seasons. He helped Zamalek win two Egyptian Premier League titles to end archrivals Ahly's domination, claiming the best player and top scorer awards last term.
He hasn't slowed down this season, scoring 10 goals and providing 10 assists in 17 games in all competitions.