So Jacob called the place Peniel, saying, "For I have seen God face to face, and yet my life has been delivered."
"But," he said, "you cannot see my face, for man shall not see me and live."
The Claim: Jacob saw God's face and lived, and this contradicts what Exodus 33:20 says.
The Explanation: This is one of those instances of taking a Bible verse out of context. In this case, it's Genesis 32:30 that is out of context. To remedy this, let's start by reading Genesis 32:22-30.
The same night he arose and took his two wives, his two female servants, and his eleven children, and crossed ford of the Jabbok. He took them and sent them across the stream, and everything else that he had. And Jacob was left alone. And a man wrestled with him until the breaking of the day. When the man saw that he did not prevail against Jacob, he touched his hip socket, and Jacob's hip was out of joint as he wrestled with him. Then he said, "Let me go, for for the day has broken."
But Jacob said, "I will not let you go unless you bless me." And he said to him, "What is your name?" And he said, "Jacob." Then he said, "Your name shall no longer be called Jacob, but Israel, for you have striven with God and with men, and have prevailed."
Then Jacob asked him, "Please tell me your name." But he said, "Why is it that you ask my name?" And there he blessed him.
So Jacob called the name of the place Peniel, saying, "For I have seen God face to face, and yet my life has been delivered."
Ok, there's a lot going on here, so let's start by addressing verse 24, which says, "And Jacob was left alone. And a MAN wrestled with him..." Clearly Jacob was wrestling with a man, and not God Almighty himself, because as Exodus 33:20 states, no man can see God's face and live. It is further evidenced that this is a man in verse 28, where the man says, "...you have striven with God AND WITH MEN..."
So who was this man that was wrestling with Jacob? He did not give his name, but this much is clear. The man was sent by God to wrestle with Jacob. So in verse 30 when Jacob says he saw God face to face, what Jacob actually saw was a man that was sent by God. The phrase "face to face" here is then just a figure of speech about a close personal encounter Jacob had, not to mean God's literal face as in Exodus 33:20. This is not a contradiction.
Extra: To take this a step further, let's examine the following question. Why did God send this man to wrestle with Jacob? In Genesis 32:31, you see that Jacob was injured and limping because of this incident, so you can see that Jacob really did get into a battle and wrestle with a man, and this wasn't some sort of dream Jacob had. So why would this happen? James 4:10 says, "Humble yourselves before the Lord, and he will exalt you." We could definitely say that this was a humbling experience for Jacob. But why does God need us to humble ourselves? We need to be humble in order to serve God and serve others. Jesus Christ humbled himself when he died on the cross for our sins. Jesus had done nothing wrong, and yet was executed. Jesus died to serve us, and Jesus would not have accepted his punishment on the cross if Jesus was not humble.