The audience is not "out to get you". They are either listening to you play because they want to enjoy your performance or they have to be there because of some sort of obligation- in which case they really aren't paying attention anyways. If you have prepared correctly and sufficiently, then identify those worries and fears about the audience for what they are: irrational and unfounded.
Most people don't want to see you fail, they are actually rooting for you. When you make a big deal about a mistake, you are actually creating a situation that is both awkward and uncomfortable for you and for them. If you just go on and don't pay the mistake any attention, the audience will forget it and forgive, and so will you. Never break your rhythm or your timing- don't apologize or make a big "oops" gesture- and learn to laugh at your mistakes.
All of the great pianists made mistakes when they played, some of them catastrophic; some of their recordings that we preserve today have some big clunkers by great musicians like Rachmaninoff and Horowitz. Arthur Rubinstein used to say that you could make a whole new piece with all of the wrong notes he played each time he played a concerto. They weren't great because they played all the notes, they were great because of their musical ideas- which were still there regardless of how many wrong notes they hit when they played. Attend a lot of live performances; perfect live playing is very rare. Realize that most of the note-perfect recordings that you have of the greats are pieced together from various takes in the studio, or are altered with the computer, or were made after many a run-through of the piece. Live recordings are usually taken from that one near perfect performance out of the nearly 300 per year that an artist gives. In a lot of cases, the artists themselves still felt they were making a compromise with the recording because the performance was not up to their standards. Annie Fischer refused to release her superb studio recording of the Beethoven sonatas while she was alive because she was never happy with them. What you listen for in what you play is wildly different from what the audience listens to. Even the different members of the audience listen for different things, as you can see when you read two different critics' takes on the same concert.
Get over the sense of shame that comes with making a mistake. That is good for improving yourself when you practice, but not good at all at the very moment of the performance. If you have a recording of a bad public performance of yourself, share it with a friend or family and laugh; an artist has to have confidence in himself when he goes out in public. You can't go in front of an audience feeling ashamed of what you do. Being cynical, most of the time people won't even notice when you make a mistake unless you stop playing, blush, repeat the passage, start over, smack your lips, frown, grumble or make some visible sign; even when the mistake was a real clunker. Are you doing all that because you are angry at your mistake, or is it because you want the audience to notice that you don't always make mistakes and that this one is so unexpected that you have to make a big deal out of it? If that is the case, then there is also a problem of arrogance or vanity that you might not be aware of.
Play in public a lot. At least five or six times before the big day. You can ease into it. Play it first for a single friend or family member and once you are comfortable start adding people. It will also give you a better idea of what needs more work and how the piece is going to feel in the actual moment. I tend to use the shock method with my students and just unexpectedly fill up my classroom with their friends to hear them run through their programme once.
Don't be afraid of your audience.