The Uruguay international has scored three goals in two appearances since serving his 10-game ban for biting Chelsea's Branislav Ivanovic.
It is early days yet but both performances have been notable for the 26-year-old's footballing brilliance and not the controversy which has dogged his time in England.
Last weekend Suarez showed his softer side by coming onto the pitch for the pre-match handshakes against Crystal Palace with new-born son Benjamin and three-year-old daughter Delfina, and he believes his family are a steadying influence for him on the field.
"I am aware that in recent matches that I played I've been calmer," he said on his return to Uruguay ahead of crucial World Cup qualifiers.
"I am very self-critical and I realised that playing well, with more tranquility, is helping me a lot.
"I realise and I prefer to continue and not be the same as before."
Suarez surprised many by appearing on the field with his children, but it is something of a tradition among South Americans and, despite initial resistance from club officials, they gave way on the striker's insistence.
"In England it is not common and the first club people told me I was not going to go with them but I told them that my children were going to come with me, like it or not," he told reporters who greeted his arrival at Montevideo airport in comments reported in Uruguayan newspaper El Pais.
"They understood in the end and it was a nice moment, a unique moment for me.
"They (family) make me think hard and calm me. Nowadays I think a lot of them when I'm on the field.
"I wanted my son to live as I do. I suffered a lot as a child and I do not want my children, or any other child, to experience the circumstances as I did.
"As a parent I try to give them all the love in the world and all the best."
Suarez dismissed some suggestions in the media that he had used his children to help win over fans on his first appearance at Anfield since the furore of his summer transfer saga when he accused the club of reneging in a deal allowing him to leave for a Champions League side.
"What the English papers say I do not care," added Suarez, who earlier in the summer had cited the British media as being one of the reasons he wanted to leave.
"The only thing I care about is playing football, and enjoying my family is what I love most."
That Suarez remained at Anfield despite interest from Arsenal, who tried to sign him with a £40,000,001 bid, owed much to the intransigence of Reds owner John W Henry, who refused to entertain selling their prize asset.
Liverpool captain Steven Gerrard admitted "waking up every morning, hoping that nothing would happen and that he would still be here" and he was very pro-active in trying to persuade the Uruguayan to stay.
It was an approach which did have some effect.
"I do not know if he prayed but what he said is what he feels because he was talking to me all the time," Suarez said.
"Gerrard, for me, is a legend in Liverpool and a great team-mate that helped me a lot.
"His attitude was an extra boost for me to take the decision to stay in Liverpool; both he and the fans of Liverpool influenced much for that.
"I admire him for the great player he is worldwide. For me he will always be a benchmark and at club level he is the best player I have played with in my career, as a person and as a footballer."