Big Belly Comedy Club received a premises licence for its proposed venue from Southwark Council in the summer but still needed planning permission to use the two empty units in the South Bank Central complex off Stamford Street.
The scheme was debated at a Southwark Council planning committee meeting on Wednesday night.
"The proposed development is located in a vibrant area of Southwark within the South Bank strategic cultural quarter, where a comedy club use is considered appropriate for the local area," senior planning officer Lara Davison told the committee.
The council received 81 objections to the planning application, mostly from residents of Rennie Court and River Court.
"The capacity of the venue at 500 people is too large for a residential area," Rennie Court resident Mike told councillors. "We've been advised that there will be up to five shows a night at weekends."
His neighbour Grace added: "This area is not Soho and it's not Covent Garden."
She continued: "The venue exits into Upper Ground which is a small one-lane road ... this area cannot handle the traffic [and] late night noise from this type of venue."
Calum Dron of Starwood Capital Group - owners of the South Bank Central complex - addressed the committee in support of the scheme, noting that Southwark as a borough currently has no dedicated stand-up comedy venue.
Michael Moucharafieh of Big Belly Comedy said: "The whole concept of Big Belly is to enjoy your night out at leisure and not to rush people in and rush people out.
"That's why we've got two smaller venues instead of one big room, so shows can be staggered."
Mr Moucharafieh insisted that there would be no DJ-led events at the venue and - contrary to residents' fears - it wouldn't become a nightclub.
He explained the two performance spaces would have capacities of 98 and 220 people, so the overall 499 capacity limit of the premises would be unlikely to be reached.
A condition of the venue's licence is that it must promote its address as 30 Stamford Street (rather than Upper Ground) to encourage patrons to arrive and depart via the main road.
Ward councillor David Watson addressed the committee to speak on behalf of his constituents.
"Borough & Bankside is absolutely a vibrant destination for visitors, but it is also home to thousands of residents, and I think the council has a duty to balance those interests.
"Residents are resoundingly of the view that this is the wrong venue in the wrong place."
Cllr Watson said that despite efforts to direct patrons to Stamford Street and away from Upper Ground he was concerned about a potential "traffic nightmare" of taxis and Ubers on that street.
Debating the application, Cllr Richard Livingstone recognised residents' concerns but welcomed the addition of a comedy venue to Southwark's cultural scene.
Cllr Richard Leeming underlined the need to support the night-time economy and cultural sectors in the borough, and welcomed the assurances given by Big Belly about how they would address the impact of their operations on neighbours.
The application won the unanimous backing of the four councillors sitting on the committee.