Nina Gross said Alice's death was a personal tragedy that shouldn't be used for political agenda, with her outrage echoed by many on social media, who said they were "disgusted" by the move.
The BBC show on Thursday night discussed the immigration issues arising from the 14-year-old's murder, after Alice's body was discovered concealed in the River Brent two days before.
Latvian Arnis Zalkalns is a suspect in the murder case, but is feared to have fled to his home country. He served seven years in prison in Latvia for killing his wife, but it appears UK authorities had no record of his conviction.
[Immigration discussion begins at 32 minutes in]
Alice's sister Nina Gross described the TV debate as "extremely insensitive" and "horrible" in a series of tweets to the programme.
She wrote: "It is extremely insensitive to use my family's tragedy for political agendas and discussion. This is a time of grief for our family. In future, please respect our wishes as we grieve.
"Now is not the time for these discussions."
In other posts, she added: "It is really insensitive and horrible that you have used our family's tragedy on Question Time."
Question Time apologised through a tweet from the show's Twitter account.
#bbcqt it is extremely insensitive to use my families tragedy for political agendas and discussion. This is a time of grief for our family.— Nina Gross (@GrossNina) October 2, 2014
"This is a personal tragedy which we want to deal with privately, rather than fearing anyone using it to for any political agenda. #bbcqt— Nina Gross (@GrossNina) October 2, 2014
@GrossNina Dear Nina, we're sorry to hear this. We're really sorry for any hurt or offence caused by tonight's programme— BBC Question Time (@bbcquestiontime) October 3, 2014
Nina Gross replied: "Thank you."
After the hunt for Alice came to a tragic end this week, her sister's Twitter biography now reads: "Love you always Alice".
In the Question Time section, host David Dimbleby introduced a question from a woman in the audience, about the "hideous murder of Alice Gross".
The question was "whether there should be freedom of movement including convicted criminals across EU borders."
Other online comments called Question Time "deplorable" and "cruel" for choosing to address the political implication of the case so soon after the tragic discovery of Alice's body.
Suggest a correction
@GrossNina So very cruel, My heart and thoughts are with you and your family. Your Beautiful sister should not be used as a political tool— Deborah Playford (@ladyfreebird750) October 2, 2014