The Uruguayan was banned for eight matches for racially abusing Evra in October but refused to shake the Frenchman's hand during Saturday's pre-match Premier League ritual, prompting widespread condemnation.
"I have spoken with the manager since the game at Old Trafford and I realise I got things wrong," Suarez said in a statement issued by Liverpool.
"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."
United manager Sir Alex Ferguson branded the Uruguayan a "disgrace" while Liverpool boss Kenny Dalglish, who has backed his striker throughout the saga, gave a robust post-game interview where he argued with a television reporter.
In another statement issued on Sunday, Dalglish said: "It is right that Luis Suarez has now apologised for what happened at Old Trafford.
"To be honest, I was shocked to hear that the player had not shaken hands having been told earlier in the week that he would do.
"When I went on TV after yesterday's game I hadn't seen what had happened, but I did not conduct myself in a way befitting of a Liverpool manager during that interview and I'd like to apologise for that."
As the club tried to limit the damage after a raft of criticism from players, fans and the media, Liverpool Managing Director Ian Ayre said in a third statement that Suarez had been wrong.
"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," said Ayre.
"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."
The match itself, between two of England's great rivals, was a tense affair and scuffles took place in the tunnel at half-time and full-time.
United accepted Liverpool's apologies in their own statement.
"Manchester United thanks Liverpool for the apologies issued following Saturday's game," it read.
"Everyone at Old Trafford wants to move on from this. The history of our two great clubs is one of success and rivalry unparalleled in British football. That should be the focus in the future of all those who love the clubs."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.