On Saturday, Reds striker Suarez refused to shake hands with Evra when the two came face-to-face for the first time since the Uruguayan was handed an eight-match ban for racially abusing the defender.
United went onto win the match 2-1, with Suarez getting one back for Liverpool late on, but fury sparked once more after an altercation inside the tunnel at full-time in which the sides clashed.
Following the match, Liverpool and Suarez issued an apology for the handshake snub, with the striker admitting his behaviour was wrong and Giggs hopes that the reaction to the game can help move the matter on.
He told reporters: “I think everyone involved wanted to move forward and we had hoped that would happen on Saturday.
"It obviously didn’t, but then 24 hours later, with the statements issued by Liverpool on Sunday, it looks like it might be the end of it.
“That’s what both clubs need and what football in general needs. The clubs need to move on. The rivalry will always be there, but let’s have it as healthy rivalry and nothing else.
“The fans were chanting about me winning more titles than Steven Gerrard on Saturday, but that’s what you get between fans and it’s what you want - you want good, healthy banter between fans.”
The idea of a pre-match handshake is now in the balance but 38-year-old Giggs believes even if the tradition is scrapped, players will still proceed with the action at some point.
“It’s only this season where it [handshake] has been a problem.” Giggs told reporters.
“If you ask most players, they’d take it or leave it. It’s just part of the build-up. It’s what you do and you are used to doing it.
“But if they take it away, so what? Players would still shake hands after the game.”
Giggs had been happy to shake Suarez’s hand even after the issue with Evra but the legendary midfielder concedes he was not sure what had happened.
He said: “I knew something had gone on because there was a bit of a kerfuffle, but I didn’t know what.
“It wasn’t until afterwards, when you saw it on the television, that you realised what had happened.”