The Manchester United manager slammed the Uruguayan's behavior before kickoff, insisting that he "could have caused a riot" given the tensions between the two clubs
Two goals from Wayne Rooney, before a late consolation from Suarez, handed United a 2-1 victory in a match that was overshadowed by the tension between Suarez and Evra.
"I couldn't believe it," Sir Alex told Sky Sports following the match. "I just could not believe it.
"I had a chat with Patrice this morning, and he said 'I'm going to shake his hand. I've got nothing to be ashamed of and I want to keep my dignity.' And then he [Suarez] refuses - he's a disgrace to Liverpool Football Club.
"That certain player should not be allowed to play for Liverpool again. With the history that club has got, and he does that in this city today - he could have caused a riot.
"I was really disappointed in that guy. It was terrible, what he did.
"It created a tension. You've seen the referee, he didn't know what to do about it. It caught him off guard, and it was a terrible start to the game. There was a terrible atmosphere created."
Sir Alex added that the debate over the wisdom of pre-match handshakes should not be allowed to hide the issue of racism in the game that has arisen this season.
"That's a different issue altogether. That started when the Premier League started, of course, and it's never been a problem really until this season, with regards to racism. It's never been a problem.
"So really we better get our house in order in terms of fighting racism. It's an issue, it's absolutely an important issue in this country. Football's come a long way from the days of John Barnes and throwing bananas to where we are today. We can't go back.
"We have to go forward, in a positive way, and ban it altogether."
The United boss did admit, however, that Evra shouldn't have celebrated in front of Suarez at the end of the match.
"He shouldn't have done that," Sir Alex said.