The pair clashed before kickoff at Old Trafford after Suarez's race case earlier in the campaign at Anfield.
"It was a very difficult time," the Frenchman recalls. "For my family as well. It was difficult, so difficult.
"I think that game at Old Trafford was the most pressure I've ever felt in my life... all about shaking his hand. For me it was very difficult before the game, to decide to shake his hand.
"I think many people would never have tried to shake his hand. But I tried and he refused. For me, it was like, 'What's going on?'.
"But I couldn't go mad. Although, you see the first tackle, a tackle on Rio Ferdinand [smiles]. If I had caught Suarez then I could have had a red card straight away and I'd have been off the pitch."
United won the match and Evra was criticised for his celebrations in front of Suarez.
"People thought my celebration at the end of the Liverpool game was because Suarez had abused me," Evra reveals. "But it was more than that. Finally I could say thanks to my team-mates.
"I know everybody was playing to win for Manchester United that day, but also a little bit for myself. That's why my celebration was like that.
"What's happened to me - personal things - this year have been really tough. But I just kept playing.
"In your life you have good and bad experiences but the most important thing is how you react, and I think I reacted in a good way. That's why I'm proud of myself.
"People can criticise me and say many things but this year I am stronger than ever, mentally. It was a big challenge this year, a personal challenge with myself."