The third-tier side, known as the Black Jaguars in the ABC Motsepe League in KwaZulu-Natal, flew out of the blocks and showed the Urban Warriors no respect by coming twice from behind as the game finished 2-2 after 120 minutes.
Menzo's side couldn't hold their nerve in the penalty shootout as Erick Chipeta hit the underside of the crossbar and Prince Nxumalo blazed over to see Ajax lose 4-3.
The former Dutch goalkeeper then went straight down the tunnel after the final whistle and the media had to camp outside the team’s change-room.
Menzo, whose side in 2015 clobbered Mainstay United 9-1 in the last 32, has, along with players, apologised for crashing out at hands of power league opposition.
“All of us, the staff, we have to feel sorry for the whole club because it was embarrassing,” Menzo told reporters after the game.
“I don't want to blame one person, I don't want to blame players … I blame myself, and everyone who was here, because we all disappointed how Ajax played to our fans and the club.”
Menzo, whose side are yet to get a win in 2017 after six games in all competitions, admitted KwaDukuza were right in their faces and gave credit to them for showing no sign of trepidation against his side.
“It's normal [that they get motivated against top teams]. You know that when a lower league team play a higher league team they always give something more and that we knew,” he added.
“And that is what they showed and big compliments to them and nothing wrong with their performance, it was great.
“But this was a chance [for us] to win the game but it is [now] a moment to get confident and win the next game and get out of this, and that is shame.”
Menzo was furious about a decision to brandish only a yellow card instead of a red when Prince Nxumalo was played clean through on goal by Abel Mabaso, remonstrating to the fourth official after referee Bafana Mosia’s decision.
However, he did not use that as an excuse and said referees also make mistakes as they are human.
“I don't know what the referee was telling me [after I approached him] about the new rules,” revealed Menzo.
“He said [Nxumalo] couldn't get the ball and the ball was far away. I think he started to pull him on the shirt at the centre line and I was not really convinced by the referee.
“They make mistakes and we make mistakes like humans so today was clear he made some mistakes, but he's human.”
The Urban Warriors will now solely focus their attention on trying to pull away from teams in the relegation zone as they lie 12th on the league table, four points clear of Highlands Park who are second from bottom.