The measures, which were announced by First Minister Mark Drakeford last month and are set to come into force on Monday (11 October), were agreed with 28 politicians voting for and 27 voting against.
However, the Night Time Industries Association (NTIA) has dubbed the vote a 'shambles' and called for an 'urgent revote' after one Conservative MP was unable to take part, with the Tories citing 'technical difficulties' for what happened.
If all opposition MS (Members of the Senedd) had taken part, the Labour-led Government would have lost in a tie, with the Tories, Lib Dems and Plaid Cymru opposed to the scheme.
In a statement the NTIA, which represents 100 venues in Wales, said: "The NTIA has long campaigned against the introduction of Covid Passports due to the operational barriers these measures will cause. We were pleased to see the number of MSs that listened to our concerns today and voted against these proposals.
"It is a democratic outrage that one MS who wanted to vote, and who would have voted against the proposals, could not, due to a mere technical error. This shambles will cause even more uncertainty for our businesses.
"There must be an urgent revote so that the will of the Senedd can be fairly expressed and businesses have some clarity about the future."
As reported by the BBC, Presiding Officer Elin Jones later said in a statement: "I gave every opportunity for the member to be present - including providing ICT support - but the member was unable to be contacted.
"For members to vote in the Senedd, they must be present, either in the chamber or on Zoom.
"It is a member's responsibility to give themselves sufficient time to secure their Zoom connection in time for voting, just as it is for any member travelling to the Senedd to vote."
Unlike the so-called 'vaccine passports' being introduced in Scotland, the NHS Covid Pass will allow adults to prove they have had a recent negative lateral flow test if they have not been fully vaccinated.
All over-18s in Wales will need to have a NHS Covid Pass to enter nightclubs; indoor, non-seated events for more than 500 people, such as concerts or conventions; outdoor non-seated events for more than 4,000 people; and any setting or event with more than 10,000 people in attendance.