Boar Home Dining, on Lower York Street in the city, will be allowed to sell booze and open after 11pm.
Selling a wide variety of hot food, it recently opened on the site of an old chippy behind Lightwaves Leisure Centre.
It runs as a takeaway as well as allowing customers to eat inside.
Two local councillors, Stuart Heptinstall and Olivia Rowley, had objected to its plan to sell alcohol, citing the area's problems with drink and drugs.
In a written statement ahead of a hearing on Tuesday, Councillor Heptinstall said the children's playground next to the premises was effectively being used as a "beer garden".
But speaking at the hearing itself, Councillor Heptinstall said he took a "balanced view" of the application.
He said: "My concern is for people living in the area and they need protection when they're out and about.
"They don't need to be intimidated by these individuals on these streets, who are on record with the police and the council.
"But on the other hand, we do need employment in the area.
"I've spoken to someone I know very well, and whose judgement I trust, and he tells me he believes these two gentlemen (running Boar Dining) can make a go of it."
Asked if the problems in the area were caused by a lack of policing, Coun Heptinstall simply replied, "Yes."
Representing the diner, Grzegorz Mateja, said he'd had experience of selling alcohol previously and had had no issues.
He also said being able to serve booze would be crucial to the success of the business, which employs five members of staff.
He told the hearing: "The prices of the alcohol I'm selling will be higher (than elsewhere in the area).
"The people causing the problems won't be coming to me for it."
Police had not objected to the application after the restaurant agreed to implement a number of "stringent" conditions.
A panel of three councillors took just 20 minutes to decide the licence should be granted.
In their reasoning, the panel said the concerns around anti-social behaviour were "understandable", but that they believed the diner would have a "positive impact on the area."
Local Democracy Reporting Service