A post on the pro-Brexit group's social media accounts said that the claims were "ill-judged and untrue".
The group had been sued by Ms Shah, who was previously a shadow minister for women and equalities and is now shadow minister for community cohesion.
The case related to a post on the Leave.EU Facebook page, made on 16 December 2019 - just days after the general election on 12 December.
The apology said: "On 16 December 2019 we posted a Facebook post which described Naz Shah MP as a 'grooming gangs apologist'.
"While the publication had been intended to be satirical; we appreciate that it was ill-judged and untrue.
"We accept that Ms Shah is not an 'apologist' for grooming gangs but is a vociferous campaigner for victims of grooming gangs.
"We apologise for the distress caused by it and have paid Ms Shah damages for libel."
The initial post was taken down on the same day it was posted, but had already been widely shared.
Ms Shah said she welcomed the apology but that screenshots of the libel were still being circulated.
She said: "I'm one of the most abused MPs on Twitter because of this allegation.
"It's horrible, it's really horrible.
"The allegation of me being a grooming apologist has been horrendous.
"The fact they've accepted that I'm a vociferous campaigner against grooming is important to me.
"This is the kind of stuff that's used against people in public life and other MPs.
"It's just not fair. It's just not right for it to happen."
Ms Shah added that she had been an prominent campaigner against child sexual exploitation.
She said: "Islamic communities are absolutely appalled by it (child sexual exploitation). People I talk to are appalled by it.
"I'm on record talking about, I've talked about it on Asian media channels and to the community. It's not something I've shied away from at all and I don't intend to in the future."
Ms Shah added that she could not talk about the damages paid, and that she was happy with the apology.
"It was never about anything else," she added.