Nearly 1,000 people attended the event at the Holy Trinity Church, Nailsea.
Afterwards, a complaint was posted on a community website by a woman "horrified" the band had "set up" on her "beloved Gran's grave".
Church rector, the Reverend Jolyon Trickey, said: "We apologise unreservedly and regret any offence caused".
The event featured live music culminating in a service and the lighting of a Jubilee beacon.
But two days after the event a woman, who has not been named, posted a complaint on the website Nailsea People.
She said that although she "had enjoyed the festivities" she had been "mortified" when she saw where the band "had set up".
"I'm still unsure as to how on earth these people got permission to set their amps up and use my Gran's grave as a stage," she said.
"She [Gran] passed away 18 months ago.
"And our children were very upset to see a man and his band dancing and singing on her grave, as were we."
Mr Trickey said he was "very happy to repeat publicly the apology already offered privately".
"An appropriate open area of grass in the Churchyard was chosen as a suitable place from which I was able to lead a short service for a crowd of nearly 1,000," he said.
"In the dark afterwards, and in their enthusiasm to entertain, the music group strayed forward onto ashes tablets under the wall.
"We failed to foresee or prevent this and have learnt lessons for the future.
"We apologise unreservedly and regret any offence caused."