Jurgen Klopp’s side, who have won 10 of their last 12 matches against Villa, climbed to third in the table on 10 points from four games. Villa have six points.
“We did quite OK, quick to 2-0 and came out good and scored a goal in the second half too,” Szoboszlai told the BBC.
Two days after Liverpool reportedly rejected a £150 million bid for Salah by Al Ittihad in Saudi Arabia, he netted goal number 188 of his Liverpool career with a toe poke at the back post in the 55th minute off Darwin Núñez‘s headed flick from a corner.
“You know it is football everyone is talking,” Szoboszlai said on Salah’s future. “We are really happy that he has stayed. 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.”
Klopp, who managed his 300th Liverpool game on Sunday, shot down rumours around Salah’s departure before kick-off.
“This was a media story and it never came true,” he told Sky Sports. “I have a big desk in my office with a lot of papers on there, I have a laptop, iPad, everything that you need and none of them are busy with this kind of news.”
Szoboszlai had given Liverpool a dream start with a left-footed half-volley from the edge of the 18-yard box after three minutes.
“It is a special moment,” he said. “If you score your first goal for Liverpool at Anfield it is always special. I am looking forward to getting even more.
“We knew if we come out and we think the game is over we were going to be in trouble but we wanted to score another one quick,” he added. “We came out and score another and just enjoy it.”
Liverpool doubled their lead when Nunez’s firm shot hit the post and rebounded into the net off Villa defender Matty Cash.
Liverpool never let up, helped by a surprisingly sloppy performance from Villa, who are now winless in their last six league visits to Anfield.
Nunez had a couple of scoring including one moments before half-time that hit the woodwork, and Trent Alexander-Arnold gave a master class in midfield before limping off with a potential injury in the second half, receiving a hug from Klopp on his way.
“We were excited before the match but the first half was key,” Villa manager Unai Emery said. “We created good chances but were not clinical and the defence was little bit weak, and they were clinical.”