The 22-year-old midfielder joined Villa on loan on 1 September, with the deal set to become permanent for thirty-five million pounds once he reaches ten appearances under Unai Emery.
Elliott had struggled for minutes under new Liverpool head coach Arne Slot, playing just seven minutes across the opening four Premier League fixtures before sealing his move.
Reflecting on his exit, Elliott told the Liverpool Echo: "It was very difficult. First of all the decision to leave and then on top of that to find the best club. I loved every minute at Liverpool. I wouldn't change it, I learned so much and experienced so many things, but I think the most important thing for me now is playing first-team football.
"Sometimes in football you need to be selfish."
He added: "The love I have for Liverpool will always be there, but I made the decision to leave because I wanted to do what's best for me. Hopefully I can kick on at Villa now."
Elliott made his Villa debut last weekend, coming off the bench for 19 minutes in the goalless draw against Everton at Goodison Park. Emery, whose side have failed to score in their opening four league matches, is expected to use the former England under-21 international to inject creativity and attacking threat into a struggling midfield.
Speaking about his new arrival, Emery said: "Harvey is a young player with quality and hunger. He has come here to play and to improve, and we will give him that opportunity."
Liverpool supporters have shown their backing for Elliott on social media, praising the midfielder for the class with which he handled his departure. The former no.19 leaves Anfield with 98 senior appearances to his name and is remembered for his energy and commitment despite never fully breaking into the starting XI.