The Uruguayan, making just his second start for the Reds following his eight-game ban for racially abusing the Frenchman, caused a ruckus between the players before kick-off after refusing to offer his hand.
Evans insisted the whole of the United team were prepared to draw a line under the previous incident at Anfield if the striker had proceeded as normal.
He told reporters: “We had been talking about it in the changing-room and whether he [Evra] would shake his [Suarez’s] hand,” said Evans.
“Myself and a few other players said to Patrice ‘just shake his hand, because you have nothing to be ashamed of or hide’. I said ‘I think if you offer a handshake then you’ll come out with a lot of respect’.
“It’s disappointing he [Suarez] didn’t shake his hand because Patrice was the one who has been racially abused.
"I think it could have ended there and then with a handshake. I actually didn’t know Suarez had refused the handshake because I was a bit further down the line.
“I didn’t know what had gone on but I heard a bit of crowd reaction. There was a bit of animosity created over the handshake, other players noticed it.”
Following United’s 2-1 victory, Evra celebrated in animated fashion in front of the United fans and Suarez, something which attracted the ire of opposing players.
And Evans admits the full-back may have been over the top but claims he was fully within his rights to behave in that manner.
He continued: “Patrice was enjoying the victory at the end and rightly so, because he’s been through a lot.
“Maybe he shouldn’t have gone to the supporters in front of Suarez, but that was his way of showing he was happy we got three points. Patrice is a great lad. Everybody loves him. He gets on with everyone and everyone loves him.
“I don’t think it was a case of doing it for Patrice. It was doing it for Patrice for what the whole club has been through and especially because of the rivalry.”
United defender Rio Ferdinand also stated he lost ‘respect’ for Suarez after he failed to shake Evra’s hand and pulled out of the handshake with the Uruguayan himself after seeing the incident.
He said: “I thought it was bad decision-making from their guy.
“It’s a touchy subject at the minute and things could have been put to bed a little bit easier if the handshake had been done.
“After seeing what I saw I decided not to shake his hand. He’s not got the respect he needs to have in these situations and acknowledge he’s made a mistake and say sorry.
“I lost all respect for the guy after that. I expected more from the other guy but it wasn’t to be.”