The Hungarian midfielder bagged his first Liverpool goal to start the scoring at Anfield, before Matty Cash turned into his net and Salah made it three.
The win was the Reds' most convincing of the 2023/24 season so far, with Jurgen Klopp's side often at their fluid, attacking best without really being troubled by Villa.
The victory – Liverpool's third from four in the Premier League so far – marked their first since the close of the summer transfer window.
Though the transfer deadline has now passed for European clubs, the window remains open in Saudi Arabia, with speculation still hovering above Salah.
Klopp has remained adamant that Liverpool's record Premier League goalscorer will not be sold for any price, as rumours swirl that Al Ittihad could make a bid worth up to £200m in order to test the reality of the Reds' public stance.
One report has even claimed that Salah's agent, Ramy Abbas, met with the Saudi champions' representatives in Dubai earlier this year and that forward's head was turned by the proposed salary package of £1.5m a week on offer .
However, speaking after the win over Villa, Szoboszlai claimed that Salah wants to remain a Liverpool player.
"It is football, everyone is talking. We are really happy that he has stayed," Szoboszlai told reporters, as quoted by AFP.
"We are of course speaking between each other but he wants to stay, he wants to be here and be with us.
"We are really happy. We need people in the team like him."
Meanwhile, Klopp revealed that while Salah has not directly confided his desire to stay, he remains committed to Liverpool.
He said (as quoted by the Guardian): “He didn’t tell me but he didn’t have to. He speaks with his training and performances and behaviour. We had meetings this week, and the meetings were not about what we did in the past, it was about what we will do in the future.
"Mo was with the players’ [leadership] committee and had his moments where he was talking and it was nothing like ‘by the way, this is only until next week’ or whatever. He is completely here and if Dom said that, fine. Mo doesn’t have to come into my office and tell me ‘by the way, boss … [I’m not going]’. For me it wasn’t a subject for one second, to be honest, besides the questions.”