Santa Montefiore said she hoped her sister knew "how much we miss you" and that she was "aware of all the lovely things people have written about you".
Ms Palmer-Tomkinson was laid to rest at a private funeral having died from a perforated ulcer at the age of 45.
Ms Montefiore later shared the eulogy, which was addressed directly to her sister, on social media.
In it the author said they had been "a family of five, but now we are four and that just isn't right.
"I hope you're aware of all the lovely things people have written about you, and that you are touched, because you never believed you lit up a room or made people feel special, but you did. You really did."
She also said she hoped her sister was "resting, because you found life hard and now it is done."
She added that she hoped "you can make rabbit ears and witty rhymes up there, because your loopy humour was your gift to us and now it is your gift to them.
"How they will laugh. I envy them that," she said.
Ms Palmer-Tomkinson rose to fame in the 1990s as a hard-partying "It girl". She received treatment in 2016 for a non-malignant brain tumour.
She was found dead at her flat in South Kensington on 8 February.
The death of Tara Palmer-Tomkinson "just isn't right", her sister said in a eulogy at the socialite's funeral.