The 25-year-old Uruguayan recently returned to action following an eight-game ban after being charged with racially abusing the Frenchman during the reverse fixture at Anfield back in October.
As the two teams lined-up for the ceremonial handshake prior to kick-off, Suarez caused further controversy when he snubbed Evra's hand - setting the tone for an eventful game that was ultimately marred by a number of unsavoury scenes.
As well as the handshake controversy, the two sets of players reportedly clashed in the tunnel at half-time before Liverpool players reacted angrily to Evra's over-zealous celebrations after the full-time whistle.
"I have spoken with the manager since the game at Old Trafford and I realise I got things wrong," read a statement from Suarez on the Liverpool 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."
Reds boss Kenny Dalglish has also made a public apology for the way in which he acted during his post-match interview in the aftermath of the game, and stated that Suarez was right to say he was sorry.
"Ian Ayre [Liverpool Managing Director] has made the club's position absolutely clear and it is right that Luis Suarez has now apologised for what happened at Old Trafford," he said.
"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. But as Ian said earlier, all of us have a responsibility to represent this club in a fit and proper manner and that applies equally to me as Liverpool manager.
"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."