Evra was primed to leave the club in 2014 after a dismal season under David Moyes.
The former France international, who confirmed his retirement on Monday, had finished his contract and was expected to pursue a new challenge elsewhere.
But he revealed vice-chairman Woodward went back on a previous agreement.
"Before the end of that season Ed told me: 'Patrice, you're going to stay for another year because we've got an option in your contract,' but I told him I needed to leave because of family reasons," he told Sky Sports.
"He shook my hand and said he understood.
"Then in May I was having dinner in Dubai on my birthday and had a text from my agent.
"He told me to go somewhere quiet and sent me a message with the statement from United saying they were pleased to renew my contract for another year. I went mad, I was really disappointed.
"I called Ed and swore on the phone, even threatened him.
"He said I couldn't talk to the director of football like that and that he was going to fine me.
"After that Ryan Giggs called and said I couldn't leave just because of one man, but it gave me an excuse to leave.
"I'd already said to my wife we were going to leave, but inside I didn't ever really feel I could. That gave me a little push.
"Some people won't understand why Ed did it but I understand.
"Of the senior players I was the only one he wanted to keep - even David Moyes had said it before he got sacked.
"They said to [Rio] Ferdinand and [Nemanja] Vidic, we don't need you anymore, but in the dressing room you need a character.
"I understood Ed, but I didn't understand why he betrayed me. He should've called me to explain.
"Without that it's difficult to know what would've happened.
"I would've had to deal with my wife - and I call her Roy Keane when she gets angry. I don't know.
"When we reached the Champions League final with Juventus she said she'd told me so, but even winning three Champions Leagues with Juve wouldn't have had the same satisfaction of playing for United."