To stage major events outside football matches, the club needs a licence which controls the sale of alcohol, how much noise is permitted, and timings. Under the terms, these types of extra events may only take place six times a year.
Councillor Portia Fahey, who represents the neighbouring Everton North ward, said the area around Anfield was "predominantly residential", describing the current restrictions as a "simple rule that works for residents".
She said in future the scheme could work but felt "the structures aren't adequate currently".
Resident Darryl Shellard, of Tancred Road, also offered an objection, claiming concert events were causing a disruption "bigger than matchdays" due to more road closures for concert-goers. He said residents were "worth more than 31 days" offered by the club.
Councillors voted by majority to back the plans to change the permissions.
It will also see the club making a commitment to donate to a community grant fund each time a concert is held.
It is understood about £140,000 has already been granted to date.