The Uruguayan, who was banned for eight matches after racially abusing Evra during the league game between the sides at Anfield in October, had been expected to shake hands with the Frenchman but failed to do so.
Suarez has apologised for his actions and Liverpool managing director Ian Ayre has now also released a statement on Sunday afternoon criticising Suarez's behaviour.
He told www.liverpoolfc.tv: "We are extremely disappointed Luis Suarez did not shake hands with Patrice Evra before yesterday's game. The player had told us beforehand that he would, but then chose not to do so.
"He was wrong to mislead us and wrong not to offer his hand to Patrice Evra. He has not only let himself down, but also Kenny Dalglish, his team-mates and the club.
"It has been made absolutely clear to Luis Suarez that his behaviour was not acceptable.
"Luis Suarez has now apologised for his actions which was the right thing to do. However, all of us have a duty to behave in a responsible manner and we hope that he now understands what is expected of anyone representing Liverpool Football Club."
Suarez issued his own apology earlier on Sunday. "I have spoken with the manager (Kenny Dalglish) since the game at Old Trafford and I realise I got things wrong," he said in a statement on the Liverpool official website.
"I've not only let him down, but also the club and what it stands for and I'm sorry. I made a mistake and I regret what happened.
"I should have shaken Patrice Evra's hand before the game and I want to apologise for my actions.
"I would like to put this whole issue behind me and concentrate on playing football."
Suarez scored in a 2-1 defeat for Liverpool, with Evra appearing to taunt the South American by celebrating right next to him at the end of the match.
United boss Sir Alex Ferguson described Suarez as "a disgrace" and suggested he should never play for Liverpool again while Professional Footballers' Association chief executive Gordon Taylor branded Suarez's conduct as "disrespectful, inappropriate and embarrassing".
Taylor added: "If anything, Patrice Evra was the victim and he was prepared to put his hand out.
"These players are expected to be role models but if we have a situation where nobody accepts the findings of hearings and just carries on regardless, all you get is anarchy.
"Now the Football Association have to step in because the whole situation has gone too far.
"Suarez had a chance to put everything to be yesterday, in front of a worldwide audience. The fact that he chose not to is, quite frankly, depressing."
Having already defended his man in a flash TV interview, Dalglish opted not to make himself available for further discussion on the matter.
Long-time friend Alan Hansen did not say on Match of the Day on Saturday night whether he had spoken to Dalglish prior to transmission.
However, his own observation was withering in its condemnation.
"Liverpool FC made a statement that there would be a handshake," he said.
"Luis Suarez was party to the statement. He knew about it and obviously agreed with it. To snub Patrice Evra was, in my view, totally unacceptable.
"The football club and the manager have given him total and unequivocal support through thick and thin.
"He has let Kenny Dalglish down, the club down and himself down."